Singularities of Solutions to Chemotaxis Systems 9783110599534, 9783110597899

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Piotr Biler Singularities of Solutions to Chemotaxis Systems

De Gruyter Series in Mathematics and Life Sciences

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Edited by Alexandra V. Antoniouk, Kyiv, Ukraine Roderick V. Nicolas Melnik, Waterloo, Ontario, Canada

Volume 6

Piotr Biler

Singularities of Solutions to Chemotaxis Systems |

Mathematics Subject Classification 2010 35Q92, 35B40, 35B44, 35B51, 35K55, 92C17 Author Prof. Dr. Piotr Biler University of Wrocław Mathematical Institute Pl. Grunwaldzki 2/4 50-384 Wrocław Poland [email protected]

ISBN 978-3-11-059789-9 e-ISBN (PDF) 978-3-11-059953-4 e-ISBN (EPUB) 978-3-11-059862-9 ISSN 2195-5530 Library of Congress Control Number: 2019951090 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Munich/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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To my beloved wife Dorota and son Andrzej

Contents Preface | IX Introduction | XI 1 The 8π-problem in the two dimensional case | XIV 2 The parabolic-elliptic Keller–Segel model in higher dimensions | XVIII 3 Chemotaxis models with nonlocal diffusion operators | XIX 4 Notations | XX 5 Preliminaries on semigroups for linear diffusion equations | XXI 1 1.1 1.2 1.3 1.4 1.5 1.6 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6

Local-in-time solutions, small global-in-time solutions | 1 Classical Keller–Segel system | 1 Fractional Keller–Segel model and optimal initial data | 6 Singular stationary solution | 6 Local and global solutions with initial data in d Morrey space M /α (ℝd ) | 9 Other simple constructions of solutions in nearly optimal spaces | 13 Notes and complements | 19 Large global-in-time solutions to models of chemotaxis | 21 The case of classical diffusion | 21 The case of a nonlocal model of chemotaxis | 30 Statement of results for α ∈ (0, 2) | 33 Pointwise comparison principle | 35 Averaged comparison principle | 43 Construction of global-in-time solutions for α ∈ (0, 1) | 49 Construction of the unique global-in-time solutions for α ∈ (1, 2) | 55 Large global-in-time solutions of the parabolic-parabolic Keller–Segel system on the plane | 58 Comments on special solutions | 66 Blowups | 73 Solutions blowing up in a finite time | 73 A novel approach to blowup and concentration of mass in two dimensions | 85 Proof of blowup of radial solutions | 89 Blowup in the plane ℝ2 with 8π concentration of mass | 91 Refinements of the proof of blowup of radial solutions in higher dimensions | 104 Blowup of solutions for the two dimensional fractional diffusion Keller–Segel model | 109

VIII | Contents 3.7 3.8

Keller–Segel model with chemoattractant consumption terms | 115 Blowup of radially symmetric solutions, again | 131

4 4.1

Blowups à la Fujita | 137 Fujita’s idea of the proof of blowup for the system with the classical diffusion | 137 Blowup à la Fujita of solutions of system with fractional diffusion | 145

4.2 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Interpretations, complements, conjectures, et cetera | 153 Dichotomy: local–global existence of solutions | 153 Further comments on solvability of the classical and fractional Keller–Segel system | 154 Optimal initial data for two dimensional Keller–Segel system | 157 Blowup of solutions to a general diffusive aggregation model | 165 Hypercontractivity of the linearization around the singular solution | 177 Classical diffusion case: super- and subsolutions | 181 Miscellanea, questions and conjectures, research plans | 188

Bibliography | 191 Index | 203

Preface The topic of this book is on the crossroad of two of my main research topics: nonlocal mean field-type nonlinear systems and equations with nonlocal diffusion. The motivations for the first topic stem from mathematical physics, in particular electrochemistry and astrophysics. The interested readers will find a panorama of recent results together with (rather technical) proofs of our (mine and my collaborators) original ones. This is a kind of warning: this book is not an introductory text to chemotaxis theory, but rather a collection of new results around global-in-time existence questions versus finite time blowup phenomena for the basic Keller–Segel model. I tried to arrange separate sections in a manner that they can be read independently, so some repetitions are unavoidable. I am much indebted to my Wrocław colleagues; my teacher and advisor, Andrzej Krzywicki, and my friend, Tadeusz Nadzieja, for introducing me to the topic of mean field models in statistical mechanics. The second topic arouse from probabilistic considerations and owes much to Wojbor Woyczyński, with whom we (Grzegorz Karch and me) studied various drift-diffusion equations related to Lévy processes in Cleveland. It is my honor and pleasure to thank all my friends for fruitful collaboration on chemotaxis over years: Waldemar Hebisch, Jacek Dziubański, Danielle Hilhorst, Andrzej Raczyński, Philippe Laurençot, Jean Dolbeault, Maria J. Esteban, Marco Cannone, Ignacio Guerra, Robert Stańczy, Peter A. Markowich, Lorenzo Brandolese, Lucilla Corrias, Gang Wu, Eduardo Elio Espejo, Noriko Mizoguchi, Jacek Zienkiewicz, Tomasz Cieślak, Dominika Pilarczyk, Xiaoxin Zheng (in a chronological order of starting our joint research), with the special affection for Grzegorz Karch. I am also indebted to Jan Burczak, Miguel A. Herrero, Cyril Imbert, Nikos Kavallaris, Changxing Miao, Régis Monneau, Yūki Naito, Takayoshi Ogawa, Michał Olech, Łukasz Paszkowski, Benoît Perthame, Takasi Senba, Mikołaj Sierżȩ ga, Takashi Suzuki, Juan-José L. Velázquez, Hiroshi Wakui, Gershon Wolansky, and Dariusz Wrzosek for the influence (direct as well as indirect) on other mathematical problems studied. I also warmly thank Tadeusz Nadzieja for pertinent remarks he offered me after reading a preliminary version of the text. Special thanks are due to Agnieszka Hejna for her help in graphics presentation. This book would never appear without the encouragement of De Gruyter Acquisitions Editor Mathematics Apostolos Damialis, who convinced me that the topic is suitable for the series “Mathematics and Life Sciences”. Last but not least, support of NCN (Polish National Center for Research; recently through the 2016/23/B/ST1/00434 project) and other grant agencies (in France, Chile, United States, Austria) is gratefully acknowledged for various projects, including a couple of small, but quite efficient bilateral French–Polish “Polonium” ones we had. Wrocław August 2019 https://doi.org/10.1515/9783110599534-201

Piotr Biler

Introduction The purpose of this book is to present recent results on chemotaxis with linear diffusion (both local and nonlocal) and quadratic nonlinear nonlocal terms described by the classical Keller–Segel model, with special emphasis on radially symmetric solutions, their short-time existence, long-time existence, asymptotics and, most importantly, blowups. We discuss some intriguing properties of solutions of the following Cauchy problem in space dimensions d ≥ 2, x ∈ ℝd , ut − Δu + ∇ ⋅ (u∇v) = 0,

Δv + u = 0,

t > 0,

t > 0,

u(x, 0) = u0 (x),

(1) (2) (3)

as well as its generalizations with nonlocal diffusion operators (14)–(15). There are many motivations to study this seemingly simple diffusion-transport mode with nonlocal and nonlinear transport term. One of them comes from Mathematical Biology, where equations (1)–(2) appear as a simplified Keller–Segel(-Patlak) (the, so called, minimal KS) system modeling chemotaxis, see the seminal paper [154], [18, 80, 102, 172, 188, 187, 208], books [210, 209] and overviews [8, 141, 142, 140, 22, 217]. The unknown variables u = u(x, t) and v = v(x, t) denote the density of the population of microorganisms (for example, swimming bacteria or slime mold), and the density of the chemical secreted by themselves that attracts them and makes them to aggregate, respectively. Another and even earlier important interpretation of system (1)–(2) stems from Astrophysics, where the unknown function u = u(x, t) is the density of gravitationally interacting massive particles in a cloud (of molecules, stars, nebulae, et cetera), and v = v(x, t) is the Newtonian potential (“mean field”) of the mass distribution u, see [94, 97, 99, 175, 225, 249, 16, 17, 15, 39, 53], and a review paper [96], also for the case of nonlinear diffusion. System (1)–(2) is one of many possible evolution models, which include steady states, such as (4), discovered by S. Chandrasekhar in [94], as a timeindependent solution. These models are related to the microscopic description of the dynamics of particle systems in kinetic theory as they appear as integrated (over velocities) versions of kinetic equations. For further studies on mathematical modeling in chemotaxis, kinetic, and mean field descriptions, see [201, 235, 234, 4, 119, 120, 204, 205, 77, 93, 175, 75, 78] and [31, 32] for models with the evolution of temperature (the so called Streater models). These parabolic-elliptic and parabolic-parabolic systems feature cross-diffusion. The cross-diffusive terms like ∇ ⋅ (u∇v) lead to substantial difficulties compared to “standard” theory of parabolic systems. Note that similar mean field models, with the + (positive) sign in equation (2) replaced by the − (negative) sign have been used for more than one century to model mihttps://doi.org/10.1515/9783110599534-202

XII | Introduction gration of electrically charged particles in electrolytes, plasma and semiconductors, see [109, 216] and [38, 30] for further references. Nonlocal diffusion operators have been studied in this context in [133]. Needless to say, biological and physical applications of equations (1)–(2) force them being considered in a domain Ω ⊂ ℝd , supplemented with relevant boundary conditions. Among the most studied boundary conditions are either the homogeneous Neumann for both u and v: 𝜕u 𝜕v = =0 𝜕ν 𝜕ν or no-flux condition

𝜕u 𝜕v +u = 0, 𝜕ν 𝜕ν together with a boundary condition (Dirichlet, Robin, et cetera) for v. Each of them formally guarantees conservation of mass ∫Ω u(x, t) dx in time. The theory of initialboundary value problems for systems is much more complicated than studies of the Cauchy problem. In particular, when the Neumann conditions are imposed on the boundary of the domain, the equation (2) for the chemoattractant spreading is re1 placed either by Δv − v + u = 0 or Δv + u − |Ω| ∫Ω u = 0 in order to determine v in a consistent way. However, certain phenomena related to local behavior of solutions (including formation of singularities) can be studied in a simpler yet substantial way for solutions of the Cauchy problem. The initial data (3) are usually nonnegative integrable functions u0 ∈ L1 (ℝd ). The total mass M = ∫ u0 (x) dx = ∫ u(x, t) dx ∈ [0, ∞) is conserved during the evolution. The conservation of mass property formally always holds, but it is not immediate in this case, and must be proved in a separate way using the current definition of the solution in each case considered. Further, we will also consider solutions with infinite mass, such as the famous Chandrasekhar steady state singular solution in [94] related to black holes in ℝd , d ≥ 3, 2(d − 2) , (4) uC (x) = |x|2 see also its generalization (1.3.1) to the case of fractional diffusion in Section 1.3. In fact, a more realistic description of chemotaxis phenomena for biologists, besides introducing boundary conditions, needs more complicated models, involving nonlinear diffusion, growth and reaction terms, and nonlinear sensitivity functions, such as ut = ∇ ⋅ (D(u)∇u − u∇χ(φ)) + g(u, φ),

εφt = Δφ + u + h(u, φ), where φ ≡ v and, for instance ε ≥ 0,

D(u) = duν ,

D(u) ≍ u(1 + uλ ),

Introduction

χ(φ) = c log φ,

φ , c+φ h(u, φ) = −γφ, . . .

χ(φ) = cφκ ,

g(u, φ) = μu(1 − u);

| XIII

χ(φ) =

Note that a popular choice of the sensitivity function χ(φ) = c log φ (the, so called, Weber–Fechner law of stimulus perception in physiology) leads to a series of pure mathematical interesting questions, see [19, 66, 88, 169, 246]. Similarly, the choice of nonlinear diffusion is interesting when balance between diffusion and transport terms is studied, see [70, 105, 144, 203, 240] for a few references. The interaction between cross-diffusion and logistic kinetics (g(u, φ) = νu−μuκ ) is an intensively studied topic since this plays an important role in population dynamics, see [245, 84] and (many) references therein. Another class of biologically relevant models appear with the following specific choice of nonlinearities for the first equation: ut = ∇ ⋅ (D(u)∇u − uh(u)∇φ), where D(s) = h(s) − sh󸀠 (s). The literature on these subjects is abundant and fast growing. For instance, Zentralblatt für Mathematik shows about 2500 entries with “chemotaxis”, and Mathematical Reviews more than 1500, and 700 with “Keller–Segel”. There are also studies on chemotaxis models coupled with equations of continuous media (they describe situation when cells and signal substance are assumed to be transported by a fluid), for example, with the Navier–Stokes system, see for instance [158, 242, 244, 66, 103, 258], and chemotaxis with attractive-repulsive interactions, for example, [255] as well as [42], reported in Chapter 5, Section 5.4. Among other modifications of the Keller–Segel model, there are multicomponent systems with quite intriguing mathematical issues (see [224] on generalizations of the Moser–Trudinger inequality, which plays an important role in the analysis of various entropy functionals, such as (5) and [250]), as well as different behaviors of solutions: [122, 123, 127, 33, 36]. Recently, chemotaxis models with an indirect signal production (with two chemicals involved) have been studied in [124, 232]. These studies are related to various functional inequalities extending the Moser–Trudinger inequality. Our goal is to study linear diffusion cases (both Brownian—local and Lévy-type— nonlocal), and quadratic nonlocal nonlinearity. We will concentrate on the simplified system (1)–(2), which deserves a deep analysis by mathematicians since even this relatively simple model features many interesting behaviors of solutions that are frequently shared by more complicated systems. The system (1)–(2) has a variational structure, so that the quantity (of a clear physical origin of entropy or free energy) W(t) = ∫ u log u dx −

1 ∫ uφ dx 2

(5)

XIV | Introduction is a Lyapunov functional decreasing along the solutions in time, because the entropy dissipation relation d W = − ∫ u|∇ log u − ∇φ|2 dx ≤ 0 dt holds. However, in majority of our results presented here, we had not used that subtle property in their proofs, unlike the authors of [7, 69, 67, 43, 44], and for the uniqueness of solutions in [92]. We adopt here most frequently the approach via mild solutions to the evolution Keller–Segel system, as it has been done in [173] for the Navier–Stokes system, but of course, the notion of weak solution is also applicable to these equations, see [38, 62]. Whenever general properties of parabolic equations are considered, [166] is used as a standard reference. Finally, it should be noted that similar phenomena take place and can be proved for nonnegative solutions of the nonlinear heat equation ut = Δu + u|u|p−1

(6)

as well as its counterparts with nonlocal diffusion operators ut + (−Δ) /2 u = u|u|p−1 α

(7)

in [1, 95, 128, 229]. Here, a general reference is the monograph [213], and some recent results can be found in [226, 21, 23, 61]; see also Chapter 5, Section 5.7.1.

1 The 8π-problem in the two dimensional case Let us now describe previous results, which motivated us to start these studies; we limit ourselves to those publications, which are directly related to that topic. We begin with the classical case of d = 2, where the value M = 8π of mass plays a crucial role. Namely, if u0 is a nonnegative measure of mass M < 8π (the subcritical case), then there exists a unique solution, which is global-in-time and bounded, see [7, 69, 64, 243] for different approaches (functional inequalities, moments, monotonicity formulas, et cetera), and its asymptotics is essentially selfsimilar in space-time. These results have been known previously for radially symmetric initial data, see [43, 44, 20, 50, 51] for recent presentations. On the other hand, if M > 8π (the supercritical case), then this solution cannot be continued to a global-in-time regular one, and a finite time blowup occurs, which means that lim sup u(x, t) = ∞ for some

t↗T, x∈ℝd

0 < T < ∞.

Introduction

| XV

The first proof of blowup was in [148], then [54, 39, 15, 20, 195, 163] appeared, and constructions of blowing up radial solutions have been presented in [137, 138]. The radial blowup is accompanied by the concentration of mass equal to 8π at the origin. Such a definition of blowup is general enough and applies to situations when the regularizing effect of the diffusion lead to bounded, hence classical solutions, see [233]. However, when mild solutions are considered, they are usually in the space of (weakly) continuous functions 𝒞w ([0, T), 𝒳 ) with values in a functional space 𝒳 on ℝd . In cases when 𝒳 contains unbounded functions and u(t) does not necessarily belong d to L∞ (ℝd ) (or when merely u(t) ∈ L∞ loc (ℝ )), it is reasonable to extend the definition of blowup to lim sup sup u(x, t) = ∞ t↗T

|x| 0, see Chapter 4, the end of the proof of Theorem 4.1.1. The book [229] is devoted to a fine description of solutions of the two dimensional Keller–Segel model at the blowup time. In particular, there is a quantization of mass property presented: the local singularities of blowing up solutions eventually grow to integer multiples of 8π. There is a few explicit examples of such blowing up solutions, see [182, 219, 238, 98]. These phenomena are closely related to the question of short time solvability of the Cauchy problem for system (1)–(2) under minimal regularity on the initial data u0 ≥ 0 in (3). Namely, if u0 is a nonnegative measure then local-in-time solution exists if and only if all the atoms of u0 have mass less than 8π, see [7], as well as Chapter 2, Section 2.2.2, Chapter 5, Section 5.3 (adopted from a recent paper [64]) for more direct and much simpler argument. A new approach to the instantaneous regularization property of solutions in some Keller–Segel systems is in [248]. The critical case M = 8π is rich in fine asymptotic behavior results [67, 68, 85, 129, 221], even in the radially symmetric case. And the case of a ball [152, 43] is quite different from the case of the whole plane in [44]. The study of radially symmetric solutions of system (1)–(2) can be reduced to a single nonlinear equation, which is no longer nonlocal Mt = 4 s Mss +

1 M Ms π

(8)

for the nondecreasing radial distribution function (or the integrated density) M defined for the density u M(r, t) = ∫ u(x, t) dx,

(9)

{|x| 0,

all of them having the property of a slow approach to the 8π level ∞

∫ (8π − Mb (s)) ds = ∞

for b > 0.

0

They are locally asymptotically stable, which has been shown by considering relative entropy functionals 𝒲b (M) = ∫(M log

M + (8π − M) log 8π − M8π − Mb ) ds. Mb

Namely, if 𝒲b (M0 ) < ∞,

then

󵄩 󵄩 lim 󵄩󵄩M(t) − Mb 󵄩󵄩󵄩L1 (0,R) = 0

t→∞󵄩

for each R > 0,

as studied in [44]. But the global dynamics picture is much more complicated; there are solutions, which diffuse the whole mass to infinity so that M(r, t) → 0 as t → ∞, solutions with an infinite time blowup that concentrate at the origin: M(r, t) → 8π for all r > 0 as t → ∞, and solutions that oscillate (“bounce”) between two different steady states Mb , see [199]. These are, in a sense, generic examples of long-time asymptotics of solutions in the critical case M = 8π; see [180, 181, 196] and Remark 5.7.4. This shows that equation (8) has a particularly rich structure of large time asymptotic behaviors of solutions. The doubly parabolic case of the Keller–Segel system, that is, when equation (1) is supplemented with the linear parabolic diffusion equation for v εvt = Δv + u

(12)

Introduction

| XVII

instead of (2), is even more difficult to study, especially when blowup questions are considered, see [190] for the two dimensional case. A striking difference of its behavior is, for example, the result in [29], obtained with the use of elementary ordinary differential equations method, and then studied in [106]. Namely, selfsimilar solutions satisfying scaling property u(x, t) = t −1 U(

x ), √t

v(x, t) = V(

x ), √t

(13)

exist for each 0 ≤ M < Mε with the optimal value of Mε = 8π whenever 0 ≤ ε ≤ ε−1 1 , whereas limε→∞ Mε = ∞. In fact, Mε ≥ max{8π, 4π }, see [29, Theorem 4]. In 2 e log ε particular, selfsimilar solutions for the parabolic-elliptic system (1)–(2) exist exactly in the range M ∈ [0, 8π). These are solutions of the Cauchy problem (1)–(12) with u0 = Mδ0 , nonunique if M > 8π. An analogous phenomenon occurs for the nonlinear heat equation as reported in [213, Theorem 20.5]. One may say that diffusion dissipates atoms less than 8π of initial measures for system (1)–(2), whereas diffusion in the doubly parabolic system (1)–(12) is able to dissipate even bigger atoms when ε is big enough. This is also a manifestation of a duality between admissible rough initial data and appearance of singularities of solutions at the blowup time, see Chapter 5, Section 5.3. We stress on the fact that the occurrence and, a fortiori, structure of singularities for the doubly parabolic system is far from being understood, see [190]. The parabolic-elliptic Keller–Segel model is the (singular perturbation) limit case as ε → 0 of the doubly parabolic system, see [214], [172] and [202] for more detail. The questions of blowup are, however, much harder for the fully parabolic version. Since there is no result with the moment approach, solely energy arguments have been used to show blowup of solutions of the doubly parabolic systems, see [190]. There is a striking result in Chapter 2, Section 2.8 on the existence of global-in-time solutions for an arbitrary u0 ∈ ℳ(ℝ2 ) and ε ≥ ε(u0 ), see [37]. This means that sufficiently slow diffusion of the chemoattractant prevents from blowup. Another asymptotic regime is studied in [191]. The role of consumption term γv in the modified equation (12) for the chemoattractant, εvt = Δv − γv + u, is discussed in Chapter 3, Section 3.7 for two dimensional doubly parabolic model (together with the dependence on diffusivity coefficient ε > 0, see [37]) and in Chapter 3, Section 3.7.2 (see [28]) in the parabolic-elliptic case (ε = 0). Namely, for each initial condition, there is ε0 = ε(u0 ) such that for ε ≥ ε0 (γ0 = γ(u0 ) and for γ ≥ γ0 , respectively) solution with u0 as the initial datum is global-in-time. On the other hand, big and well concentrated initial conditions still lead to blowups, see Chapter 3, Section 3.6.

XVIII | Introduction

2 The parabolic-elliptic Keller–Segel model in higher dimensions In view of results in the two dimensional case mentioned above (when the parameter of the total mass M plays decisive role in the temporal behavior of solutions), for d ≥ 3, ̃ 0 ) that determines the blowup. More prewe are looking for a critical quantity ℓ̃ = ℓ(u ̃ 0 ) < c(d) implies global-in-time cisely, do constants 0 < c(d) ≤ C(d) exist such that ℓ(u ̃ 0 ) > C(d) leads to a finite time blowup of existence of solution to (1)–(2), whereas ℓ(u solution? These critical quantities are, of course, related to norms of function spaces that are scale invariant for the systems considered, such as Lebesgue space L1 (ℝ2 ) and finite d Radon measures on ℝ2 ([16, 7, 64]), Lebesgue space L /2 (ℝd ) ([159, 107]), Marcinkiewicz d d d d/2 Lw (ℝd ) = L /2,∞ (ℝd ) = L /2,∗ (ℝd ) ([161]), Morrey Mq/2 (ℝd ) ([16, 18, 172], as well as more general Morrey-type spaces for the doubly parabolic Keller–Segel system in [239]), Besov ([63, 90, 251, 256, 257]), Fourier–Herz spaces ([252]) and Fourier–Besov–Morrey ([101]), as well as more exotic pseudomeasures space 𝒫ℳa (ℝd ) = {v ∈ 𝒮 󸀠 (ℝd ) : v̂ ∈ L1loc (ℝd ), ‖v‖𝒫ℳa ≡ sup |ξ |a |v̂(ξ )| < ∞}, a = 2 (in [27]). Of course, the role of scaling invariants is well known for various evolution equations, see for instance [149, 90]. We will give an answer to the above-mentioned dichotomy question in Corollary 4.1.9, showing that ℓ̃ is close to the radial concentration defined below in (19), d (20)—and thus equivalent to the Morrey norm in the space M /2 (ℝd ). A generalization α to the case of the dissipation defined by a fractional power of Laplacian (−Δ) /2 with α ∈ (1, 2) is in Chapter 3, Section 3.6 and Chapter 4, Section 4.2 (see [65]), showing that d ℓ̃ is close to the Morrey norm in the space M /α (ℝd ), see Chapter 4, Section 4.2. Various equivalent conditions for blowup in ℝd , d ≥ 3, are summarized in Remark 4.1.8. In that way, we may say that in higher dimensional case interesting behaviors d arise around the critical value 2σd of the quantity (equivalent to the Morrey M /2 (ℝd ) space norm for radial functions) lim R2−d ∫ u0 (x) dx,

R→∞

{|x| 0,

u(x, 0) = u0 (x).

(14) (15) (16)

General references to analytic theory of Lévy operators and their applications can be found in [147, 179, 165, 228]. Probabilistic motivations to study stochastic processes relative to systems, whose trajectories have jumps (for example, motion of swimming bacteria alluded to in the Introduction) have been described in [34, 35, 48, 49, 45, 40, 62] and references therein. Time evolutions of probability distribution functions associated with such stochastic processes are described by deterministic equations studied here. We quote, among many relevant references, papers [1] (on nonlinear equations with general integral operators describing diffusion), [128], [95] (on linear diffusion equations and large time asymptotics of their solutions). Closely related to chemotaxis are [121, 76, 82–84]. Well-posedness (and ill-posedness) have been treated in [63, 111, 145, 172, 251, 252]. Qualitative properties of chemotaxis models with nonlocal diffusion have been studied in [28, 41, 51, 52, 176, 230, 167, 241]. There is a popular belief among PDE theorists that existence of solutions for an equation is fundamental, but “easy” (even if not easy, this is a first step when we try to build a model of real life phenomena), but nonexistence of (global, regular) solutions is somewhat more intricate and “difficult”. Indeed, the nonexistence of an object signals that the considered model is either too crude or leads to a phenomenon, which was beyond the consideration when the model had been set up. For the existence— there are usually many possible and well-established approaches (classical, weak, mild, renormalized, et cetera solutions), whereas for nonexistence, there is no “canonical” way to study this. Following that paradigm, we consider in this book existence and nonexistence questions of global regular solutions, looking for certain quantities (preferably, functional norms) that distinguish data leading to “nice” solutions from those “bad” ones leading to formation of singularities. A deep problem (yet connected with the determination of admissible, but not regular enough initial data) emerges: What is the asymptotic profile of blowing up solution? That is, does blowup occur with a fixed singularity (here ≍ |x|1 2 ), or with a concentration of mass as in the fundamental examples in d = 3 of Herrero, Medina, Velázquez in [136, 135, 137–139]? And, what is the blowup rate of ‖u(t)‖ as t ↗ T? Some new promising results for the Keller–Segel model in a ball in this direction are in [227].

XX | Introduction In the following Chapters we will discuss existence of local-in-time solutions and small global-in-time solutions of the considered systems, 2: existence of large global-in-time solutions to models of chemotaxis, 3: various approaches to show blowup of solutions, 4: Fujita’s approach to blowup, 5: complementary remarks. 1:

4 Notations Let us recall some relevant definitions of functional norms. In the sequel, standard notation, ‖ ⋅ ‖p is used to denote the Lebesgue Lp (ℝd ) norm; ‖ ⋅ ‖W s,p denotes the Sobolev space W s,p (ℝd ) norm; H s = W s,2 . ℳ(ℝ2 ) denotes the Banach space of finite Radon measures on ℝ2 with the usual total variation norm, and the weak convergence tested with all continuous compactly supported functions φ ∈ 𝒞0 (ℝ2 ). Integrals with no integration limits are meant to be calculated over the whole space ℝd , and C’s are generic constants independent of t, u, . . . which may, however, vary from line to line. The homogeneous Morrey spaces Mqs (ℝd ) modeled on Lq (ℝd ), q ≥ 1, are defined for 1 ≤ q ≤ s < ∞ by their norms Rd(q/s−1)

|u||Mqs ≡ ( sup

R>0, x∈ℝd

=

sup

R>0, x∈ℝd

1/q

∫ {|y−x|0, x∈ℝd

Rd(1/s−1/q) ‖1B(x,R) u‖Lq,∗ < ∞.

(18)

Caution: The notation for Morrey spaces used elsewhere might be different, for instance Mqs is denoted by M q,λ with λ = dq/s and by Ṁ sq in [172]. For various facts related to function spaces, we refer the readers to [3], [184], and for another approach to [237]. For α ∈ (1, 2], the dα -radial concentration of a locally integrable radial function u ≥ 0 in this book is defined by ||u|| d = sup Rα−d ∫ u(y) dy. α

R>0

{|y|0

(20)

{|y|δ}

u(x − y) − u(x) dy, |y|d+α

(23)

where, by [115, Theorem 1], [165], 𝒜 = 𝒜(d, α) =

2α Γ( d+α ) 2

π d/2 |Γ(− α2 )|

.

(24)

XXII | Introduction For many important identities related to the fractional Laplacian, we refer the reader to [81, 116, 165], and the Lévy–Khintchine representation for (23) is in (3.1.17). This operator generates a semigroup of linear convolution operators on various α/2 p L (ℝd ) spaces, 1 ≤ p ≤ ∞, which will still be denoted as e−t(−Δ) , and from the context, it will be clear where it is defined. α/2 α/2 Recall that the integral kernel Pt,α of e−t(−Δ) satisfying e−t(−Δ) z = Pt,α ∗ z, is of selfsimilar form d

Pt,α (x) = t − /α R(

|x| ) ≥ 0. t 1/α

(25)

2

For α = 2, we have, of course, R(ϱ) = (4π)− /2 exp(− ϱ4 ), and Pt,2 is decaying exponentially. It is well known (see [155, 156] and [147, example 3.9.17]) that, for α ∈ (0, 2), the function R has an algebraic decay at infinity d

0 ≤ R(y) ≤ C(1 + |y|d+α ) ,

(26)

C , + |x|)d

(27)

−1

in other words 0 < Pt,α (x) ≤

(t 1/α

and, of course, ‖Pt,α ‖1 = 1. We will also need estimates for the kernel Ht of the gradient Ht = ∇Pt,α (x), which has the form Ht (x) = t −(d+1)/α H( with

x ), t 1/α

C 󵄨 󵄨󵄨 , 󵄨󵄨Ht (x)󵄨󵄨󵄨 ≤ 1/α (t + |x|)d+1

(28)

(29)

see [72], and satisfies the relations 1

‖Ht ‖1 = Ct − /α ,

(30)

󵄨 󵄨󵄨 −(d+1)/α , 󵄨󵄨Ht (x)󵄨󵄨󵄨 ≤ Ct 1/α−1 󵄨󵄨 󵄨󵄨 −(d+α+1) |x| . 󵄨󵄨Ht (x)󵄨󵄨 ≤ Ct

(31) (32)

It is also of interest to recall the Bochner subordination formula, see [254, Chapα/2 ter IX.11], which gives a dimension-free representation of the semigroup e−t(−Δ) using the usual heat semigroup eλΔ e

α/2

−t(−Δ)

∞

= ∫ ft,α (λ)eλΔ dλ 0

(33)

Introduction

| XXIII

with the functions ft,α independent of the dimension d and satisfying e

−tsα

∞

= ∫ ft,α (λ)eλs dλ. 0

α/2

Therefore, the kernel Pt,α of e−t(−Δ) can be expressed as ∞

2

d

Pt,α (x) = ∫ ft,α (λ)(4πλ)− /2 e−|x| /4λ dλ.

(34)

0

For some fine estimates (note that the function R decays algebraically) together with 𝜕 all its derivatives, R󸀠 , R󸀠󸀠 , . . . , 󸀠 = 𝜕ϱ : R(ϱ) ≍ ϱ−d−α , Here, r = |x| and ϱ =

R󸀠 (ϱ) ≍ ϱ−d−1−α , r 1 . In |T−t| /α −1

R󸀠󸀠 (ϱ) ≍ ϱ−d−2−α , . . .

as

ϱ → ∞.

(35)

fact, using the Fourier transform, we see that the

function R satisfies R(|x|) = ℱ (exp(−|ξ |α ))(x). This is normalized so that ∞

d−1

σd ∫ R(ϱ)ϱ

∞

σ 󵄨 󵄨 dϱ = d ∫ 󵄨󵄨󵄨R󸀠 (ϱ)󵄨󵄨󵄨ϱd dϱ = 1, d

(36)

0

0

with R(0) = (2π)−d ∫ exp(−|ξ |α ) dξ ∞

d

= (2π)−d α−1 σd ∫ e−τ τ /α−1 dτ = 0

2Γ( dα )

α(4π)d/2 Γ( d2 )

.

(37)

For other aspects of Schrödinger-type operators with fractional Laplacians and potentials, see [71, 143]. Considering the classical Brownian diffusion case, α = 2, G = Pt,2 denotes the Gauss–Weierstrass kernel of the heat semigroup etΔ on Lp (ℝd ) space d

G(x, t) = (4πt)− /2 exp(−

|x|2 ). 4t

(38)

As it is well known, the convolution with G, denoted by G(t) ∗ z = etΔ z, satisfies the following Lq − Lp estimates 󵄩󵄩 tΔ 󵄩󵄩 d(1/p−1/q)/2 ‖z‖q 󵄩󵄩e z 󵄩󵄩p ≤ Ct

(39)

󵄩󵄩 tΔ 󵄩󵄩 −1/2+d(1/p−1/q)/2 ‖z‖q 󵄩󵄩∇e z 󵄩󵄩p ≤ Ct

(40)

and

for all 1 ≤ q ≤ p ≤ ∞, t > 0.

XXIV | Introduction Here, we recall also some analytic inequalities for Morrey norms and semigroups from [233], [147, example 3.9.17], [131], similar to those for the action between the Lebesgue spaces. These are analogous to the estimates for the heat semigroup for α = 2 in [130, Proposition 3.2], [233, Theorem 3.8, (3.71)–(3.75), (4.18)] recalled in [16, (13)–(14)], and can be, for example, obtained using inequalities (30)–(32). For 1 ≤ p1 ≤ p2 ≤ ∞, α/2

|e−t(−Δ) z||M p2 ≤ Ct −d(1/p1 −1/p2 )/α|z||M p

(41)

holds. Moreover, for 1 ≤ p1 ≤ p2 ≤ ∞, the estimate for the gradient of the semigroup reads α/2

|∇e−t(−Δ) z||M p2 ≤ Ct −1/α−d(1/p1 −1/p2 )/α|z||M p1 .

(42)

A further extension to Morrey spaces with two indices is as follows: for 1 < p1 < p2 < ∞ and q2 < pp2 , 1

α/2

|e−t(−Δ) z||M p2 ≤ Ct −d(1/p1 −1/p2 )/α|z||M p

(43)

q2

holds. Moreover, for 1 < p1 < p2 < ∞, q1 > 1 and qq2 = pp2 , provided p1 ≤ d, 1 1 otherwise, the estimate for the gradient of the semigroup reads α/2

|∇e−t(−Δ) z||M p2 ≤ Ct −1/α−d(1/p1 −1/p2 )/α|z||Mqp1 . q2

1

q2 q1


0, d

x ∈ ℝ , t > 0,

u(x, 0) = u0 (x),

d

x∈ℝ .

(1.1.1) (1.1.2) (1.1.3)

A construction of weak solutions of problem (1.1.1)–(1.1.2) with sufficiently regular initial data (1.1.3), say u0 ∈ Lp (ℝd ) with p > d/2, can be done using rather standard techniques in e. g. [38, 223]. However, as we remarked in Introduction, we adopt here another strategy of construction of mild solutions based on the regularization property of the heat semigroup, and in next Sections, semigroups generated by fractional Laplacians. By a local-in-time mild solution of problem (1.1.1)–(1.1.2), we understand a measurable function u = u(x, t), which is finite almost everywhere, satisfying the Duhamel formula (the variation of parameters formula) u(t) = etΔ u0 + ℬ(u, u)(t)

(1.1.4)

with suitably defined bilinear form ℬ, see (1.1.7). Such a definition of solution permits us to consider solutions that are not necessarily bounded for large x (as shown in Chapter 4), and those with local singularities, such as the Chandrasekhar stationary solution uC . For the blowup questions, we will consider solutions that instantaneously regd ularize to L∞ loc (ℝ ), those with sufficiently strong regularization effect exerted by the α/2

semigroup (either etΔ or e−t(−Δ) ). See also Remark 5.2.2 for examples of data that lead to nonregularization phenomena. For such data as in (1.1.4), if u0 ≥ 0 is radially symmetric then, by the uniqueness, the corresponding solution u of problem (1.1.1)–(1.1.2) is also nonnegative and radial as long as it exists. It is well known that if the total mass M is finite, it is conserved during the evolution, as long as a regular solution exists for t ∈ [0, Tmax ) M = ∫ u0 (x) dx = ∫ u(x, t) dx ℝd

for all t ∈ [0, Tmax ).

(1.1.5)

ℝd

Further, we will also consider locally integrable solutions with infinite mass, such as the famous Chandrasekhar steady state singular solution (4) introduced in [94]. Recall that, due to its singularity at x = 0, solution (4) is neither weak nor distributional solution for most physically interesting cases d ∈ {3, 4}, which leads to considerable https://doi.org/10.1515/9783110599534-001

2 | 1 Local-in-time solutions, small global-in-time solutions mathematical subtleties, see for example [27]. This one-point singular solution plays a pivotal role in analysis of this system, because it allows distinguishing (in some sense) radial initial conditions of global-in-time solutions from initial data, where solutions cannot exist for all t > 0. As preliminary results on general (including sign changing and not necessarily radial) solutions we will show d – local-in-time existence of solutions with initial data in Morrey spaces M /2 (ℝd ) ∩ p d M (ℝ ), p ∈ (d/2, d) in Proposition 1.1.2 i); – global-in-time existence of solutions with sufficiently small initial data in Morrey d spaces M /2 (ℝd ) ∩ M p (ℝd ), with p ∈ (d/2, d) in Proposition 1.1.2 ii). The proofs of these auxiliary results are based on the classical Fujita–Kato iterations d procedure in suitably chosen spaces contained in the critical space M /2 (ℝd ), which p d admit local singularities only in M (ℝ ), thus weaker than the critical one of uC . The modern presentation of that technique is based on the following fixed point result for equation (1.1.4): 1 Lemma 1.1.1. If ‖ℬ(u, z)‖ℰ ≤ η‖u‖ℰ ‖z‖ℰ and ‖etΔ u0 ‖ℰ ≤ R < 4η , then equation (1.1.4) has a solution, which is unique in the ball of radius 2R in the space ℰ . Moreover, these solutions depend continuously on the initial data, that is, ‖u − u‖̃ ℰ ≤ C‖etΔ (u0 − ũ 0 )‖ℰ .

The detailed proof of Lemma 1.1.1 can be found in [172], [16]. The reasoning involves the Banach contraction theorem, the unique solution being achieved as a limit in ℰ of the sequence of successive approximations w0 (t) = etΔ u0 ,

wn+1 = w0 + ℬ(wn , wn ).

(1.1.6)

In fact, the latter result is optimal (if it concerns the minimal regularity of admissible initial data), and valid under the sole smallness assumption on the norm d of u0 ∈ M /2 (ℝd ), see [172, Theorem 1 B), C)] with a rather technical proof, involving Morrey spaces modeled on Marcinkiewicz spaces. Moreover, the Cauchy problem for d system (1.1.1)–(1.1.3) is not well posed for large data in M /2 (ℝd ), see subsection 5.2.2 in Chapter 5 Section 5.2 with α = 2. For a generalization of all these results (and abovementioned Remarks) to the Keller–Segel system with fractional diffusion, see Introduction, Section 3 (and [65]). Remark 5.2.1 deals with the initial trace of any nonnegative solution of a more general Keller–Segel system with fractional diffusion (1.2.1)–(1.2.2), and again, for α = 2, d the Morrey space M /2 (ℝd ) norm enters as a critical quantity, which measures the minimal regularity of the initial data needed for the existence of a local-in-time solution of that system. As we have already remarked in the Introduction, critical regularity and size of initial data are intimately connected with the form of singularities at the blowup time.

1.1 Classical Keller–Segel system

| 3

Proposition 1.1.2. d i) Given u0 ∈ M /2 (ℝd ) ∩ M p (ℝd ) with d ≥ 2 and p ∈ (d/2, d), there exist T = T(u0 ) > 0 and a unique local-in-time solution, u ∈ 𝒞w ([0, T], M /2 (ℝd ) ∩ M p (ℝd )) d

d

󵄩 󵄩 ∩ {u : (0, T) → L∞ (ℝd ) : sup t 2p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩∞ < ∞} 0 0. d ii) Moreover, if u0 ∈ M /2 (ℝd ) is sufficiently small, then the solution is global-in-time d/2 u ∈ 𝒞w ([0, ∞), M (ℝd )), enjoys the time decay property supt>0 t β|u(t)||M r < ∞ for r ∈ (d, ∞] and β = d2 ( p1 − 1r ), as well as interior 𝒞 ∞ -regularity. Remark 1.1.3. Note that uC ∈ M /2 (ℝd ) and |uC|M d/2 = ||uC || = 2σd . The second assumpd

tion u0 ∈ M p (ℝd ), with p > d/2, is a kind of regularity assumption, which rules out local singularities of the strength |x|1 2 . Indeed, 1{|x|R} uC ∈

M p (ℝd ) with p > d/2.

Remark 1.1.4. Note that, in general, we have only weak convergence of etΔ u0 to an initial datum either u0 ∈ M p (ℝd ), 1 ≤ p ≤ ∞, or u0 ∈ Lp (ℝd ), p ∈ {1, ∞}, whereas u(t) is norm continuous for t ∈ (0, T). Thus, we are obliged to consider weakly continuous d 𝒞w ([0, T], M /2 (ℝd ) ∩ M p (ℝd )) instead of the space of norm continuous (denoted by 𝒞 ) functions of time variable t with values in a Morrey/Lebesgue space. Proof of Proposition 1.1.2. We sketch the proof of this result noting that the reasonings are, in a sense, close to those in [16, Prop. 1, Th. 1]. We consider the Duhamel integral equation (1.1.4) with the bilinear form ℬ defined by t

ℬ(u, w)(t) = − ∫ ∇e

(t−s)Δ

(u∇Ed ∗ w)(s) ds,

(1.1.7)

0

since we choose ∇v = ∇Ed ∗ u as the solution of equation (1.1.2) with the fundamental solution Ed of the Laplacian −Δ in ℝd . Consider an auxiliary space d/2

d

p

d

𝒳 ≡ 𝒞w ([0, T], M (ℝ ) ∩ M (ℝ ))

d 󵄩 󵄩 ∩ {u : (0, T) → L∞ (ℝd ) : sup t 2p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩∞ < ∞}

0 d/α, is a kind of regularity assumption, which rules out local singularities stronger than or equal to |x|1 α . This also have its counterpart for global-in-time solutions emanating from suitably small data. Here, we assume neither radial symmetry nor nonnegativity of solutions. Proposition 1.4.2. d i) Given u0 ∈ M /α (ℝd ) ∩ M p (ℝd ) with d ≥ 2, α ∈ (1, 2], and p ∈ (d/α, d), there exist T = T(u0 ) > 0 and a unique local-in-time mild solution u ∈ 𝒳 ≡ 𝒞 ([0, T], M /α (ℝd ) ∩ M p (ℝd )) d

of problem (1.2.1)–(1.2.3), in the sense of the mild formulation (1.4.1). d ii) Moreover, if u0 ∈ M /α (ℝd ) is sufficiently small, then T can be chosen arbitrarily d large, and the solution is global-in-time: u ∈ 𝒞 ([0, ∞), M /α (ℝd )). These solutions are smooth and enjoy certain decay in time t as t → ∞, see the proof for detailed information. Proof. We sketch the proof of this result noting that the techniques are close to those in Proposition 1.1.2 (thus, [16, Proposition 1, Theory 1]) in the case α = 2, see also [46, Proposition 3.1] for the case α = 2. Consider an auxiliary space r

d

𝒴 ≡ {u : (0, T) → M (ℝ ) :

Here, p < d < r, β = α/2

d 1 ( α p

sup t β|u(t)||M r < ∞}.

00 z(r, t) < ∞ for each t ∈ [0, Tmax ). Moreover, since u(t) ∈ M p (ℝd ) with p > d/2 and the local-in-time bound on this norm, we obtain z(r, t) ≤ r 2−d r d−d/p|u(t)||M p → 0

as r → 0

(2.1.14)

for each t ∈ [0, Tmax ). The proof of inequality (2.1.12) is by contradiction. Suppose that for some t0 > 0 and R > 0 the solution M(r, t) hits the barrier b(r) in (2.1.11) at (R, t0 ), and t0 can be chosen as the least such moment t > 0. The two parts of the graph of the 1 ϵ2σ ̃ t) = r d−d/p M(r, t) barrier b(r) meet at r∗ = ( K d ) 2−d/p . If 0 < R < r∗ , then the function z(r,

hits the constant level K at r = R and t0 . If R ≥ r∗ , then z(r, t) = r 2−d M(r, t) hits the ̃ t) < constant level 2σd at r = R and t0 . Indeed, by the assumptions either supr>0 z(r, K (perhaps with a slightly bigger K) or supr>0 z(r, 0) < 2σd . Thus, there exists some ϵ0 ∈ (0, 1) such that z(r, 0) < ϵ0 2σd for each r ≥ 0. By the local-in-time existence and continuity, z(r, t) < ϵ2σd holds for some ϵ ∈ (ϵ0 , 1) and each t ∈ [0, τ) with some τ > 0. And similarly with z.̃ Hence, by a simple argument, one may show that the function z(r, t0 ) attains its local maximum at R and 𝜕t𝜕 z(R, t0 ) ≥ 0. Of course, the same holds with z.̃ Thus, the

2.1 The case of classical diffusion

| 25

relations zr (R, t0 ) = 0,

zrr (R, t0 ) ≤ 0

(2.1.15)

hold for either z or z.̃ On the other hand, applying the relations to equation (2.1.13), we get 𝜕 d−2 2 d−2 z (R, t0 ) z(R, t0 ) = zrr (R, t0 ) − 2 2 z(R, t0 ) + 𝜕t R σd R2 d−2 ≤ z(R, t0 )(z(R, t0 ) − 2σd ) < 0 σd R2 d−2 ϵ2σd (ϵ − 1)2σd < 0, = σd R2 which is a contradiction. For the case of the function z,̃ we consider the analogous equation (r d−d/p z)̃ t = (r d−d/p z)̃ rr −

d − 1 d−d/p 1 ̃ d−d/p z)̃ r , z)̃ r + r 1−d (r d−d/p z)(r (r r σd

(2.1.16)

also equivalent to equation (2.1.8). Let us compute the time derivative of z̃ at the point (R, t0 ), analogously as was done above taking into account zr̃ (R, t0 ) = 0,

̃ (R, t0 ) ≤ 0, zrr

𝜕 (d − d/p)(d − 1) ( (d − d/p)(d − d/p − 1) ̃ t0 ) − ̃ t0 ) = zrr ̃ (R, t0 ) + z(R, z̃ R, t0 ) z(R, 2 𝜕t R R2 d − d/p 1−d ̃ t0 )Rd−d/p−1 R z(R, + σd ≤

d − d/p 1 d ̃ t0 )) < 0 ̃ t0 )(− + z(R, z(R, 2 p σd R

d − d/p 1 d ̃ t0 )) ̃ t0 )(− + R2−d/p z(R, z(R, p σd R2 d − d/p d ϵ2σd ̃ t0 )(− + = z(R, ) M = limr→∞ M0 (r) = ∫ℝd u0 (x) dx. This, in a completely analogous way again, implies M(r, t) < min{K r d−d/p , 2σd r d−2 , N}.

(2.1.18)

Under assumption (2.1.5) in Theorem 2.1.1, equation (2.1.8) leads to a differential inequality 2σ 𝜕M d−1 ≤ Mrr − Mr + d Mr 𝜕t r σd r d−3 = Mrr − Mr r ≤ Mrr

(2.1.19)

since d ≥ 3 and Mr ≥ 0. Thus, the solution of the problem (2.1.8) on (0, ∞) × (0, T), M(0, t) = 0, with the initial condition M0 satisfying for some d/2 < p < d and K > 0 M0 (r) ≤ min{Kr d−d/p , 2σd r d−2 , N},

(2.1.20)

with u0 ∈ M /2 (ℝd ) ∩ M p (ℝd ) ∩ L1 (ℝd ) and ||u0 || < 2σd , as was assumed in Proposition 1.1.2 and Proposition 2.1.7, is a subsolution of the Cauchy problem for the one dimensional heat equation on the half line d

mt = mxx ,

(x, t) ∈ (0, ∞) × (0, T),

0 ≤ m(x, 0) ≤ max{Kxd−d/p , 2σd xd−2 , N} m(0, t) = 0

for t ≥ 0,

for x > 0,

(2.1.21)

m(x, 0) = m0 (x) for x > 0. As it is well known, the solution of the initial-boundary value problem (2.1.21) can be represented (using the reflection method, the extension of solutions of (2.1.21) to odd functions on ℝ solving the heat equation) as r0

m(r, t) = (4πt)

−1/2

2

2

∫(e−(r−x) /(4t) − e−(r+x) /(4t) )m0 (x) dx.

(2.1.22)

0

Now, let us fix 0 < R1 < R2 < ∞ with small R1 and large R2 to be determined later. By the bound (2.1.18), we have for r ∈ (0, R1 ) z(r, t) ≤ K r 2−d/p ≤ KR2−d/p 1

(2.1.23)

2.1 The case of classical diffusion

| 27

since d/p ∈ (1, 2). Again by inequality (2.1.18), for r ∈ (R2 , ∞), we get z(r, t) ≤ Nr 2−d ≤ NR2−d 2 .

(2.1.24)

Finally, we obtain for r ∈ [R1 , R2 ] z(r, t) ≤ R2−d sup m(r, t) 1 r>0

≤

−1/2 NR2−d 1 (4πt)

∞

2

2

∫ (e−(r−y) /(4t) − e−(r+y) /(4t) ) dy 0

r

2

≤ NR2−d sup (4πt)−1/2 ∫ e−ρ /(4t) dρ 1 r∈[R1 ,R2 ]

−r

−1/2 sup = NR2−d 1 (4π)

− √r t

r∈[R1 ,R2 ]

≤ cNR2−d 1

1 √t

2

∫ e−ϱ /4 dϱ − √r t

for some constant c > 0.

(2.1.25)

Putting together inequalities (2.1.23), (2.1.24), (2.1.25), we arrive at lim sup z(r, t) ≤ KR2−d/p + lim cNR2−d 1 1 t→∞

t→∞

1 + NR2−d 2 √t

and the right-hand side can be done as small as we wish with a suitable choice of R1 d and R2 . Therefore, limt→∞ supr>0 z(r, t) = 0 holds whenever u0 ∈ M /2 (ℝd ) ∩ M p (ℝd ) ∩ L1 (ℝd ) and ||u0 || < 2σd . To obtain the Lq decay of solutions stated in Corollary 2.1.5, we formulate an important result on radial solutions of the Poisson equation and their gradients. Lemma 2.1.8. Let u ∈ L1loc (ℝd ) be a radially symmetric function such that v = Ed ∗ u 1 1 log |x| and Ed (x) = (d−2)σ |x|2−d for d ≥ 3 solves the Poisson equation with E2 (x) = − 2π d

Δv + u = 0. Here, the area of the unit sphere 𝕊d ⊂ ℝd is denoted by σd = Then ∇v(x) ⋅ x = −

d

2π /2

Γ( d2 )

1 2−d |x| σd

(2.1.26)

.

∫

u(y) dy.

{|y|≤|x|}

Proof. By the Gauss–Stokes theorem, we obtain for the radial distribution function M of u y dS. (2.1.27) M(R) ≡ ∫ u(y) dy = − ∫ ∇v(y) ⋅ |y| {|y|≤R}

{|y|=R}

28 | 2 Large global-in-time solutions to models of chemotaxis Thus, for the radial function ∇v(x) ⋅ ∇v(x) ⋅ x =

x |x|

and |x| = R, we arrive at the identity

1 y 1 2−d R dS = − R2−d M(R), ∫ ∇v(y) ⋅ σd |y| σd {|y|=R}

which completes the proof. Remark 2.1.9. Notice that each radial function u = u(x) satisfies for |x| = R the equality u(x) =

1 1−d 𝜕 M(R), R σd 𝜕R

(2.1.28)

which results immediately from the definition of M(R) in (2.1.27), written in the polar coordinates. Proof of Corollary 2.1.5. Under the assumption u0 ∈ M /2 (ℝd ) ∩ M p (ℝd ) ∩ Lq (ℝd ), by the proof of Proposition 1.1.2, u(t) ∈ L∞ (ℝd ) holds, so we infer u ∈ 𝒞 1 ((0, T], Lq (ℝd )). Next, let us multiply equation (2.1.1) by uq−1 with some q > 1. After integrations by parts, we obtain d

1 d ∫ uq dx = −(q − 1) ∫ |∇u|2 uq−2 dx + (q − 1) ∫ u∇v ⋅ ∇u uq−2 dx q dt q − 1 󵄨 q 󵄨2 q−1 q q = −4 2 ∫󵄨󵄨󵄨∇u /2 󵄨󵄨󵄨 dx + 2 ∫ u /2 ∇v ⋅ ∇u /2 dx q q q

≤ −4

q − 1 u /2 q − 1 󵄨󵄨 q/2 󵄨󵄨2 󵄨 q 󵄨 |x ⋅ ∇v|󵄨󵄨󵄨∇u /2 󵄨󵄨󵄨 dx. ∫󵄨󵄨∇u 󵄨󵄨 dx + 2 ∫ 2 q |x| q

In the last line, we have use the inequality |∇v| ≤ |x|−1 |x ⋅ ∇v| valid for the radial function v = v(x). Using Lemma 2.1.8, combined with the result in Corollary 2.1.3, |x ⋅ ∇v| =

1 1 2−d r M(r, t) ≤ sup z(r, t) → 0 σd σd r>0

as

t → ∞.

Next, applying the Hardy inequality (see [5])

we arrive at the estimate

2 (d − 2)2 󵄩󵄩󵄩󵄩 f 󵄩󵄩󵄩󵄩 2 󵄩󵄩 󵄩󵄩 ≤ ‖∇f ‖2 , 4 󵄩󵄩 |x| 󵄩󵄩2

(2.1.29)

d 2 1 q − 1 󵄩󵄩 q/2 󵄩󵄩2 󵄩 q 󵄩2 sup z(r, t)󵄩󵄩󵄩∇u /2 󵄩󵄩󵄩2 , ‖u‖qq ≤ −4 󵄩∇u 󵄩󵄩2 + 2(q − 1) dt q 󵄩 q − 2 σd r>0

so that for sufficiently large t > 0, when supr>0 z(r, t) is small enough by property (2.1.6), d 󵄩 q 󵄩2 ‖u‖qq + μ󵄩󵄩󵄩∇u /2 󵄩󵄩󵄩2 ≤ 0 dt

with μ > 0.

and t ≥ T, T suitably large. This concludes the proof. In fact, this is with any μ < 4 q−1 q

2.1 The case of classical diffusion

| 29

Proof of Corollary 2.1.4. Using Remark 2.1.5 and the Gagliardo–Nirenberg inequality 2 s ‖w‖2+s 2 ≤ C(d, p)‖∇w‖2 ‖w‖p

where s =

4p , d(2−p)

for

1 0, since the mass conservation property for nonnegative with s = solutions gives ‖u(t)‖1 ≤ ‖u0 ‖1 . This leads to the differential inequality of the form f 1+ /2 (t) f 󸀠 (t) ≤ −C‖u0 ‖−qs/2 1 s

for the function f (t) = ‖u(t)‖qq and C > 0, which finally gives an algebraic decay of the Lq norm 󵄩 󵄩󵄩 −d/2(1−1/q) ‖u0 ‖1 . 󵄩󵄩u(t)󵄩󵄩󵄩q ≤ Ct Remark 2.1.10. To illustrate Theorem 2.1.1, let us consider the positive radially symmetric selfsimilar solutions described in more detail in Appendix 2.9.2. They emanate from initial data, which are multiples of |x|1 2 , and do not decay as t → ∞ in the sense of their radial concentration. Indeed, they are of the form |x| 1 u(x, t) = U( ) √t t

(2.1.30)

with a profile U and u0 (x) = εuC with some (small) ε > 0. Thus, they do not satisfy assumptions of Theorem 2.1.1. According to the result in [172, Theorem 1 B), C)] mentioned in the Introduction (see also [16, Theorem 3], [18, Theorem 4]), they are small d in the space M /2 (ℝd ). See also Remark 5.7.1 on a work in progress [47] on selfsimilar solutions. The radial concentration of such a solution is z(r, t) = σd r

2−d

r

ρ 1 ∫ U( )ρd−1 dρ √t t 0

= σd (

2−d

r ) √t

r ≡ Φ( ) √t

r √t

∫ U(ϱ)ϱd−1 dϱ 0

for some function Φ, so that its supr>0 z(r, t) is time independent.

30 | 2 Large global-in-time solutions to models of chemotaxis

2.2 The case of a nonlocal model of chemotaxis This section contains mainly results reported in the recent paper [52]. 2.2.1 Formulation of the problem We consider the following version of the parabolic-elliptic Keller–Segel model of chemotaxis in d ≥ 2 space dimensions α

ut + (−Δ) /2 u + ∇ ⋅ (u∇v) = 0, Δv + u = 0,

x ∈ ℝd , t > 0, d

x ∈ ℝ , t > 0,

(2.2.1) (2.2.2)

supplemented with the nonnegative initial condition u(x, 0) = u0 (x).

(2.2.3)

Here, the unknown variables u = u(x, t) and v = v(x, t) correspond to the density of the population of microorganisms (for example, swimming bacteria or slime mold) and the density of the chemical substance secreted by themselves that attracts them and makes them to aggregate. In this work, a diffusion process described by model α (2.2.1)–(2.2.3) is given by the fractional power of the Laplacian (−Δ) /2 with α ∈ (0, 2), α which is a pseudodifferential operator with a symbol |ξ | , see (23). The initial datum in (2.2.3) is a nonnegative function u0 ∈ L1 (ℝd ) of the total mass M = ∫ℝd u0 (x) dx, which is conserved during the evolution of (suitably regular) solutions M = ∫ u(x, t) dx

for all t ∈ [0, T).

(2.2.4)

ℝd

Recall that a natural scaling for system (2.2.1)–(2.2.2) uλ (x, t) = λα u(λx, λα t)

for each λ > 0

(2.2.5)

leads to the equality ∫ℝd uλ (x, t) dx = λα−d ∫ℝd u(x, t) dx, that is, for α ≠ d, the total mass of a rescaled solution uλ can be chosen arbitrarily with suitable λ > 0. 2.2.2 Some new thoughts on the 8π-problem in the classical case Let us now describe previous results, which motivated us to begin this study. Since there is already a huge amount of literature on different models of chemotaxis, we are going to limit ourselves to those publications, which are directly related to studies in this Section. We begin with the classical case of α = 2 and d = 2, where mass M = 8π

2.2 The case of a nonlocal model of chemotaxis |

31

plays a crucial role. Namely, if u0 is a nonnegative measure of mass M < 8π, then there exists a unique solution, which is global-in-time, see [7, 69, 64]. These results have been known previously for radially symmetric initial data, see [43, 44, 20, 50] for recent presentations. On the other hand, if M > 8π, then this solution cannot be continued to a global-in-time regular one, and a finite time blowup occurs, see [15, 195, 163], and [39, 20] for radially symmetric case. The radial blowup is accompanied by the concentration of mass equal to 8π at the origin. In the general case, this concentration phenomenon occurs with a quantization of mass equal to 8kπ, k ∈ ℕ, see [231, Chapter 15].

2.2.3 Parabolic-elliptic model in higher dimensions Now, we discuss the case of α = 2 and d ≥ 3 in the model (2.2.1)–(2.2.3). It is well known that problem (2.2.1)–(2.2.3) with α = 2 has a unique local-in-time mild solution u ∈ 𝒞 ([0, T); Lp (ℝd )) for every u0 ∈ Lp (ℝd ) with p > d/2, see [16, 149, 159]. For solvability results in other functional spaces, see also [27, 26, 63, 149, 172, 230]. In particular, previous works have dealt with the existence of global-in-time solutions with small data in critical spaces, those which are scale invariant under the natural scaling (2.2.5), see [16, 27, 26, 149, 172]. Here, as usual, by definition a mild solution satisfies a suitable integral formulation (1.4.1) of the Cauchy problem (2.2.1)–(2.2.3). Due to a parabolic regularization effect, this solution is smooth for t > 0, hence, it satisfies the Cauchy problem in the classical sense. Moreover, it conserves the total mass (2.2.4) and is nonnegative when u0 ≥ 0. Proofs of these classical results can be found in [16, 160, 163, 172]. 2.2.4 Subcritical case α ∈ (1, 2) Results on local-in-time (and also global-in-time) solutions to the Cauchy problem (2.2.1)–(2.2.3) with subcritical α ∈ (1, 2) in various functional spaces (Lebesgue, Besov, Morrey) have been obtained in [62, Theorem 2.2], [63, Theorem 1.1], [41, Theorem 2.1], [172, Theorem 2] and Section 2.7 below. They are, roughly speaking, analogous to those for α = 2. Nonexistence of global-in-time solutions to problem (2.2.1)–(2.2.2) with α < 2 corresponding to large initial conditions has been proved in [41, 42, 176, 177, 178].

2.2.5 Supercritical case α ∈ (0, 1] For supercritical α ∈ (0, 1] there are results on the local-in-time solvability of (2.2.1)– (2.2.3) with the initial data in Besov spaces in [230, Theorem 1, Theorem 2, Theorem 3, Remark 10]. Other solvability results with rather smooth initial data u0 ∈ H s (ℝ2 ) ∩

32 | 2 Large global-in-time solutions to models of chemotaxis Lq (ℝ2 ), s > 3, 1 < q < 2, can be found in [178, Theorem 1.1], see also Theorem 2.3.1. Recall that (see [230, Remark 7]) if u0 ∈ L1 (ℝd ) is radially symmetric and nonnegative, then the solution constructed in [230, Theorem 1, Theorem 2] is also radially symmetric, nonnegative, and satisfies the mass conservation property (2.2.4).

2.2.6 Brief description of results for Keller–Segel system with fractional diffusion Motivated by the existence of the threshold value of mass M = 8π playing an important role in the study of problem (2.2.1)–(2.2.3) on the plane and with α = 2, we try to identify the threshold size of initial data such that corresponding solutions of problem (2.2.1)–(2.2.3) with d ≥ 2 and α ∈ (0, 2) either exist or do not exist for all t ≥ 0. In this Section, we limit ourselves to nonnegative radially symmetric solutions. The singular generalized Chandrasekhar solution of the form uC (x) = s(α, d)|x|−α , see (1.3.1)–(1.3.2), plays a crucial role in our construction of global-in-time solutions. In Theorem 2.3.1, we consider problem (2.2.1)–(2.2.3) with α ∈ (0, 1), and we assume that a nonnegative, radial, and sufficiently regular initial datum stays below the singular steady state uC (x). In this case, we are always able to construct global-in-time solutions. Global-in-time solutions of problem (2.2.1)–(2.2.3) with α ∈ (1, 2) are obtained in Theorem 2.3.3. Here, however, we have to assume that the initial datum stays below uC (x) in the following integral (averaged)—so weakened compared to pointwise— sense: ∫ u0 (x) dx < ϵ ∫ uC (x) dx = ϵ {|x| 0 such that if Rα−d 0 ∫{|x| Cα,d for some R0 > 0, then the corresponding solution of problem (2.2.1)–(2.2.3) with α ∈ (0, 2] cannot be global-in-time. Theorem 3.8.1 implies also that problem (2.2.1)– d (2.2.3) is locally ill posed in the space 𝒞 ([0, T], M /α (ℝd )), see Remark 3.8.2 for more detail. Further, in Remark 3.7.3, we try to estimate the value of the number C(α, d) and σd s(α, d) required in the construction of globalto compare it with the critical value d−α in-time solutions in Theorem 2.3.3. What happens with local-in-time solution emanating from data slightly above uC (either in the pointwise or the averaged sense), but below 𝒪(1)uC , is an open problem. Nonuniqueness of solutions of the Cauchy problem and their discontinuity with respect to the initial data can be expected for those supercritical initial data, as in the case of the classical nonlinear heat equation in [213] as well as for its extensions with nonlocal diffusion in [61]. We discuss this issue in Chapter 5, Section 5.1.

2.3 Statement of results for α ∈ (0, 2)

| 33

Anyway, the singular solution uC is identified as a threshold for the initial data separating the lower region of global-in-time solutions from the upper region of solutions blowing up in a finite time. In the next section, Section 2.3, we state and discuss all our results. Sections 2.4 and 2.5 contain the proofs of Theorem 2.4.1 (for α ∈ (0, 1)) and Theorem 2.5.1 (for α ∈ (1, 2)), asserting that if an initial datum stays below the steady state, then so is the corresponding solution. These two comparison principles allow us to construct global-in-time for α ∈ (0, 1) in Theorem 2.3.1, proved in Section 2.6, and for α ∈ (1, 2) in Theorem 2.3.3 proved in Section 2.7. One of the fundamental technical difficulties is related to explicit expressions of fractional Laplacians applied to radially symmetric functions, and their optimization. Our blowup results stated in Theorem 3.8.1 are proved in Section 3.8. Recall that the case α = 2 of classical diffusion in the Keller–Segel system is studied using slightly different methods in Chapter 2, Section 2.1. The results on the optimal conditions for global-in-time existence of radial and nonnegative solutions are also in [46].

2.3 Statement of results for α ∈ (0, 2) As we have already mentioned, the critical value of mass M = 8π decides whether a nonnegative integrable initial datum in problem (2.2.1)–(2.2.3) with α = 2 and d = 2 leads to a global-in-time solution or not. In the case of α ≠ d, mass cannot play such a role anymore due to the scaling (2.2.5). Thus, when studying a blowup phenomenon of solutions to problem (2.2.1)–(2.2.3), the following natural question arises: how to determine threshold for a size and for a singularity of an initial datum such that the corresponding solution of problem (2.2.1)–(2.2.3) is still regular and global-in-time? In this Chapter, in the series of four theorems, we partially answer this question in the case of radially symmetric nonnegative solutions of problem (2.2.1)–(2.2.3) with α ∈ (0, 2) \ {1}. We begin by emphasizing that this question is intimately related to the existence of stationary, radial, and homogeneous solution, whose exact form will play a crucial role in the statements and the proofs of our next results. In the following two theorems, we construct global-in-time radially symmetric solutions to problem (2.2.1)– (2.2.3) with α ∈ (0, 1) and α ∈ (1, 2), respectively, and with large, sufficiently regular, nonnegative initial conditions, which are below the singular steady state uC . Unfortunately, the methods presented here cannot be applied to problem (2.2.1)–(2.2.3) with α = 1. Theorem 2.3.1 (Global-in-time solutions in supercritical case). Assume α ∈ (0, 1), n = 2p > d + 1 with p ∈ ℕ, ϵ ∈ (0, 1) and K > 0 (large). There exists γ0 ∈ (0, α) (γ0 = γ0 (ϵ, α) independent of K and sufficiently close to α) and N > 0 (N sufficiently large) such that

34 | 2 Large global-in-time solutions to models of chemotaxis if any radially symmetric initial datum u0 satisfies 0 ≤ u0 (x) < min{N,

K ϵs(α, d) } , |x|γ0 |x|α

for all

x ∈ ℝd \ {0},

(2.3.1)

then problem (2.2.1)–(2.2.3) has a radially symmetric, global-in-time solution u ∈ 𝒞 ([0, ∞), W 4,n (ℝd )) ∩ 𝒞 1 ([0, ∞), W 3,n (ℝd ))

(2.3.2)

such that u(t) ∈ L1 (ℝd ) for each t > 0. Moreover, this solution satisfies the bound 0 ≤ u(x, t) < min{N,

K ϵs(α, d) } , |x|γ0 |x|α

for all

x ∈ ℝd \ {0}, t > 0.

(2.3.3)

The proof of Theorem 2.3.1 given in Section 2.6 is based on a pointwise comparison principle, involving the singular steady state uC (x) = s(α, d)|x|−α which is rather unusual property of solutions to models of chemotaxis. Note that a comparison property in the whole generality does not hold for the Keller–Segel chemotaxis systems. More precisely, we will show below in Theorem 2.4.1 that if a sufficiently regular radial initial datum satisfies estimate (2.3.1), then the corresponding solution must stay below a special barrier constructed with the use of the singular steady state uC (x). Remark 2.3.2. In Theorem 2.3.1, we consider sufficiently smooth initial data u0 ∈ W 4,n (ℝd ) with n = 2p > d + 1 and n ∈ ℕ. Thus, there exist always large N > 0 and large K > 0 such that 0 ≤ u0 (x) < min{N, K|x|−γ } for each γ ∈ (0, α). Thus, assumption (2.3.1) requires, in fact, that sufficiently smooth, nonnegative, and radial u0 is below the multiple ϵuC of the singular steady state (1.3.1) with some ϵ < 1. In our next theorem, we construct global-in-time solutions in the subcritical case α ∈ (1, 2) and with initial conditions in the homogeneous Morrey spaces M p (ℝd ). The key property is another version of the comparison principle, which is valid for integrated radial solutions. Theorem 2.3.3 (Global-in-time solutions in the subcritical case). Let α ∈ (1, 2), d > 2α, γ ∈ (0, α) (γ close to α) and ϵ ∈ (0, 1) (ϵ close to 1). Assume that the nonnegative radial initial datum u0 ∈ L∞ satisfies ∫ u0 (x) dx < min{KRd−γ , ϵ {|x| 0, t > 0,

(2.3.4)

for some K > 0 (large), where the number s(α, d) is defined in (1.3.2) in Theorem 1.3.1. Then, the corresponding solution u of system (2.2.1)–(2.2.2) is nonnegative, global-intime, and satisfies the estimates ∫ u(x, t) dx < min{KRd−γ , ϵ {|x| 0, t > 0.

2.4 Pointwise comparison principle

| 35

This theorem is proved in Section 2.7. Remark 2.3.4. Observe that if 0 ≤ u0 ∈ M p (ℝd ) for some p > 1, then, by definition (17), we have ∫{|x| 0. Thus, the estimate

∫{|x| |u0|M p . Remark 2.3.5. On the other hand, if a nonnegative radial function v = v(x) satisfies ∫ v(x) dx ≤ CRd−κ

for all R > 0

{|x| 0 and κ ∈ [1, d), then, in fact, v belongs to the Morrey space M /κ (ℝd ), see Proposition 2.7.1 for the proof. Thus, inequality (2.3.4) expresses certain assumpd d tion on u0 in terms of the norms in M /γ (ℝd ) and M /α (ℝd ). d

Remark 2.3.6. Note that for the singular stationary solution uC (x), we have the identity ∫ uC (x) dx = s(α, d) ∫ {|x| 0, there exist γ0 ∈ (0, α) (γ0 = γ0 (ϵ, α) independent of K and sufficiently close to α) and N > 0 (sufficiently large) such that every radial solution u ∈ 𝒞 1 (ℝd × [0, T]) of system (2.2.1)–(2.2.2) with the properties lim |x|α u(x, t) = 0

uniformly on [0, T],

|x|→∞

(2.4.1)

and the initial datum u0 satisfying 0 ≤ u0 (x) < min{N,

K ϵs(α, d) , } ≡ b(x) for all |x|γ0 |x|α

x ∈ ℝd

(2.4.2)

satisfies the estimate 0 ≤ u(x, t) < b(x) for all

x ∈ ℝd and 0 ≤ t ≤ T.

(2.4.3)

Besides Lemma 2.1.8, we need a technical estimate of some important constants formulated in the following: Lemma 2.4.2. For each d ∈ ℕ, d ≥ 2, and α ∈ (0, 1], the following inequality 𝒜σd ≥ αs(α, d)

(2.4.4)

holds, where 𝒜 is defined in (24), s(α, d)—in (1.3.6) and σd —in (2.1.26). Proof. We note that inequality (2.4.4) is equivalent to the following relation for the Gamma function: ) Γ( d+α 2

Γ(1 − α2 )Γ( d2 )

≥

+ 1)Γ(α) Γ( d−α 2

Γ( d2 − α + 1)Γ( α2 )

Indeed, this is an immediate consequence of the relation 𝒜σd =

) 2π d/2 2α Γ( d+α 2 , π d/2 |Γ(− α2 )| Γ( d ) 2

obtained from (24) and (2.1.26), and of αs(α, d) = α2α

Γ( d−α + 1)Γ(α) 2

Γ( d2 − α + 1)Γ( α2 )

.

(2.4.5)

| 37

2.4 Pointwise comparison principle

by (1.3.6), as well as of the property of the Gamma function: α2 |Γ(− α2 )| = Γ(1 − α2 ). Now, estimate (2.4.5) is, in turn, equivalent to the following one: Γ( d2 − α + 1)Γ( α2 )Γ( α2 ) Γ( d−α + 1)Γ(α) 2

≥

Γ( α2 )Γ(1 − α2 )Γ( d2 ) Γ( d+α ) 2

(2.4.6)

,

that is, to α α α α α d α d B( , − α + 1)B( , ) ≥ B( , 1 − )B( , ), 2 2 2 2 2 2 2 2

(2.4.7)

where B is the Euler Beta function defined as 1

Γ(μ)Γ(ν) Γ(μ + ν)

B(μ, ν) = ∫ τμ−1 (1 − τ)ν−1 dτ = 0

for all μ, ν > 0.

(2.4.8)

Clearly, for α = 1, inequality (2.4.6) is satisfied. For α ∈ (0, 1), by the Hölder inequality, we obtain 1

∫τ

α −1 2

(1 − τ)

d −1 2

1

dτ ≤ (∫ τ

α −1 2

(1 − τ)

d −α 2

1 p

dτ) (∫ τ

0

0

1

α −1 2

(1 − τ)

α −1 2

1 q

dτ) ,

0

and 1

∫τ

α −1 2

− α2

(1 − τ)

1

dτ ≤ (∫ τ

α −1 2

(1 − τ)

d −α 2

1 q

dτ) (∫ τ

0

0

1

α −1 2

(1 − τ)

α −1 2

1 p

dτ) ,

0

with p=

d 2

3α 2 d−α 2

+1

−

,

q=

d 2

−

3α 2

+1

1−α

,

1 1 + = 1, p q

since 1 − α > 0. Putting those inequalities together, we arrive at inequality (2.4.7). Now, let us begin the proof of the comparison principle in the supercritical case. Proof of Theorem 2.4.1. Let u be a solution of problem (2.2.1)–(2.2.3) for an initial datum u0 satisfying relations (2.4.1) and (2.4.2). The proof of inequality (2.4.3) is by contradiction. Suppose that there exists t0 ∈ (0, T], which is the first moment when u(x, t) hits the barrier b(x) defined in (2.4.2). By a priori 𝒞 1 regularity of u(x, t), and by property (2.4.1), the value of t0 is well defined. Moreover, there exists xt0 ∈ ℝd , satisfying u(xt0 , t0 ) = b(xt0 ). In the following, we use the numbers R∗ = (

1/γ 0

K ) N

and R# = (

1/(α−γ0 )

ϵs(α, d) ) K

,

(2.4.9)

38 | 2 Large global-in-time solutions to models of chemotaxis which are the values of R = |x| corresponding to the intersection points of three curves forming the graph of the barrier b(x). Here, we choose N so large to have 0 < R∗ < R# . We consider an auxiliary function ̃ (x, t0 ) = |x|γ u(x, t0 ), u

(2.4.10)

where the value of γ depends on |xt0 | in the following way: α { { γ = {γ0 { {0

|xt0 | ≥ R# , R∗ ≤ |xt0 | < R# , |xt0 | < R∗ .

if if if

(2.4.11)

̃ (x, t0 ) as a Here, the constant γ0 ∈ (0, α) will be chosen later on. It is easy to see that u function of x attains its local maximum at xt0 . Indeed, by the choice of γ, the function ̃ (x, t) = |x|γ u(x, t) hits the modified barrier |x|γ b(x) at a constant part of its graph. u ̃ (x1 , t0 ) > u ̃ (xt0 , t0 ) would contradict the Hence, the existence of x1 ≠ xt0 such that u choice of t0 as the first hitting point of the barrier. Thus, we have ̃ (x, t0 )|x=xt = 0. ∇u 0

(2.4.12)

Taking into account (2.2.2), equation (2.2.1) can be rewritten as ut = −(−Δ) /2 u + u2 − ∇u ⋅ ∇v. α

(2.4.13)

Here, we have the identity ̃ ) = −γ|x|−2−γ x u ̃ + |x|−γ ∇u ̃. ∇u = ∇(|x|−γ u Thus, for radially symmetric solutions, by Lemma 2.1.8 and formula (2.4.10), we get ̃ ) + |x|−γ u ̃2 − ̃ t = −|x|γ (−Δ) /2 (|x|−γ u u α

γ −d ̃ M(|x|, t) − ∇u ̃ ⋅ ∇v, |x| u σd

where the radial distribution function M is defined in (2.1.27). Hence, by equation (2.4.12), we obtain at the point x = xt0 and t = t0 𝜕 α ̃ 2 (xt0 , t0 ) ̃ )|(xt ,t0 ) + |xt0 |−γ u ̃ (xt0 , t)|t=t0 = − |xt0 |γ (−Δ) /2 (|x|−γ u u 0 𝜕t γ ̃ )(xt0 , t0 ). ̃ (xt0 , t0 )M(|xt0 |, t0 ) ≡ B(u − |xt0 |−d u σd

(2.4.14)

Our goal is to show that the right-hand side of equation (2.4.14) is strictly negative. It ̃ (xt0 , t) has to increase as a function of t in a neighwill give a contradiction because u γ borhood of t0 to hit the barrier |x| b(x) at point xt0 (recall that |x|γ b(x) is constant in a neighborhood of xt0 ) at a moment of time t0 .

2.4 Pointwise comparison principle

| 39

We begin by auxiliary results. For radial functions u = u(x, t), abusing slightly the ̃ (x, t) = u ̃ (R, t) and u ̃ (y, t) = u ̃ (r, t), where R = |x| and notation, we will simply write u r = |y|. In this new notation, we rewrite the last term on the right-hand side of (2.4.14) as follows: R

γ −d ̃ (x, t)M(|x|, t) = γR−d u ̃ (R, t) ∫ r d−1−γ u ̃ (r, t) dr. |x| u σd

(2.4.15)

0

To deal with the fractional Laplacian term in (2.4.14) with 0 < α < 1, we use the definition (23) with the constant (24). Applying formula (23) to the function ω(x) = ̃ (x, t), we arrive at u(x, t) = |x|−γ u α

̃ )(x, t) = 𝒜 P.V. ∫ −(−Δ) /2 (|x|−γ u

̃ (x, t) ̃ (x − y, t) − |x − y|γ u |x|γ u dy |x|γ |x − y|γ |y|d+α

1 1 1 − ) dy ( |y|d+α |x − y|γ |x|γ ̃ (x − y, t) − u ̃ (x, t) u dy. + 𝒜 P.V. ∫ |x − y|γ |y|d+α

̃ (x, t)𝒜 P.V. ∫ =u

(2.4.16)

Recalling the notation R = |x|, let us express the second term on the right-hand side ̃=u ̃ (x, t) in polar coordinates as follows: of (2.4.16) for radially symmetric u 𝒜 lim δ↘0

∫ {|x−y|>δ}

̃ (y, t) − u ̃ (x, t) u dy = 𝒜 lim δ↘0 |y|γ |x − y|d+α

= 𝒜 lim δ↘0

{||x|−|y||>δ}

̃ (r, t) − u ̃ (R, t)) ∫ (u

∫

𝕊d

{|r−R|>δ} R−δ

∫

̃ (y, t) − u ̃ (x, t) u dy |y|γ |x − y|d+α

dσ r d−1−γ dr |x + rσ|d+α

∞

r ̃ (r, t) − u ̃ (R, t))R−d−α ϕ( )r d−1−γ dr, = lim 𝒜( ∫ + ∫ )(u R δ↘0 0

(2.4.17)

R+δ

where the function ϕ is defined by ϕ(τ) ≡ ∫ 𝕊d

dσ > 0, |e1 + τσ|d+α

(2.4.18)

with 𝕊d denoting the unit sphere in ℝd and e1 = (1, 0, . . . , 0) ∈ ℝd . This function satisfies ϕ(0) = σd =

d

2π /2

Γ( d2 )

< ϕ(τ) for

0 0, ϵ ∈ (0, 1), and 𝒜σd ≥ αs(α, d) for each α ∈ (0, 1) (see Lemma 2.4.2), we obtain r R

𝒜ϕ( ) − αϵs(α, d) ≥ 𝒜σd − αϵs(α, d) > 0.

(2.4.24)

̃ (r, t0 ) increases with respect to r and, unHence, the integral in (2.4.23) increases as u ̃ (x, t) ≤ ϵs(α, d), its maximum is attained at a constant function der the constraint u ̃ (r, t) = u ̃ (R0 , t0 ) = ϵs(α, d). u The integrand of the second integral on the right-hand side of (2.4.21) is nonpos̃ (r, t0 ) ≤ ϵs(α, d) = u ̃ (R0 , t0 ) by the definition itive also, because of the constraint u of R0 and t0 . Its maximum equals zero and this is attained at the constant function ̃ (r, t0 ) ≡ ϵs(α, d), as well. u ̃ )(xt0 , t0 ) attains its maximum at Consequently, for each δ > 0, the quantity Bδ (u ̃ (r, t0 ) = ϵs(α, d). Now, we may pass to the limit δ ↘ 0 using the constant function u the formula in (2.4.20) to conclude that the right-hand side of equality (2.4.14) attains ̃ (x, t) ≤ ϵs(α, d)) at the constant function u ̃ (r, t) ≡ its maximum (under the constraint u α ϵs(α, d). Hence, using formulas (2.4.15) and (1.3.7) for (−Δ) /2 (|x|−α ), we have α 𝜕 2 2 α ̃ (xt0 , t)|t=t0 ≤ − Rα0 (−Δ) /2 (|x|−α ϵs(α, d))||x|=R0 + R−α u R−α 0 (ϵs(α, d)) − 0 (ϵs(α, d)) 𝜕t d−α d − 2α 2 2 α 2 −α = s(α, d) R0 (−ϵ +ϵ −ϵ ) d−α d−α d − 2α )ϵ(−1 + ϵ) < 0, = s(α, d)2 R−α 0 ( d−α where the last inequality is obtained because α ∈ (0, 1) and ϵ ∈ (0, 1). Case 2. R∗ ≤ R < R# , where R∗ and R# are defined in (2.4.9). Similarly as was in Case 1, we look for the maximum of Bδ in (2.4.21) within an extended class of admissible func0 ̃ (x, t) ≤ K. Let us first show that 𝒜σd ≥ γ0 KRα−γ , where γ0 ∈ (0, α) is arbitrary tions u 0 at this stage of the proof. Indeed, using inequalities (2.4.24) and R0 ≤ R# , we get α−γ0

𝒜σd ≥ αs(α, d) > γ0 ϵs(α, d) = γ0 KR#

α−γ0

≥ γ0 KR0

.

Hence, again as before, the first term on the right-hand side of (2.4.21) is nonnegative ̃ (x, t0 ) ≤ K. Thus, as in Case 1, passing to the in the class of functions satisfying 0 ≤ u

42 | 2 Large global-in-time solutions to models of chemotaxis ̃ (x, t0 ) ≤ K, we obtain that the constant limit δ ↘ 0, and under the constraint 0 ≤ u ̃ (x, t0 ) ≡ K maximizes the right-hand side of (2.4.20), that is, function u γK 2 −γ 𝜕 α ̃ (xt0 , t)|t=t0 ≤ −Rγ0 (−Δ) /2 (|x|−γ )󵄨󵄨󵄨󵄨|x|=R + R−γ K2 − R . u 0 0 𝜕t d−γ 0

To continue, we recall that

Rγ (−Δ) /2 (|x|−γ )(R) = R−α Cα,γ α

with Cα,γ =

d−γ α+γ )Γ( 2 ) 2 d−α−γ γ Γ( 2 )Γ( 2 )

2α Γ(

α

—a consequence of formula (1.3.8) for (−Δ) /2 (|x|−γ ). We also

α−γ

need the inequality KR0 0 ≤ ϵs(α, d), which is obvious by the definition of R# . Note also that we have the inequality −Cα,α + s(α, d)(1 −

α ) ≤ 0, d−α

because this is equivalent to estimate (2.4.4). Thus, given ϵ ∈ (0, 1) there exists γ0 ∈ (0, α), sufficiently close to α, such that γ 𝜕 K ̃ (xt0 , t)|t=t0 ≤ α (−Cα,γ + KRα−γ (1 − u )) 𝜕t R d−γ

γ K )) < 0. (−Cα,γ + ϵs(α, d)(1 − Rα d−γ

≤

1

Case 3. R0 = |xt0 | ≤ R∗ = (K/N) /γ0 . Here, by definition (2.4.11), we have γ = 0. Thus, equation (2.4.14) reduces to 𝜕 α ̃+u ̃2 . ̃ (xt0 , t)|t=t0 = −(−Δ) /2 u u 𝜕t

(2.4.25)

We estimate the right-hand side of equation (2.4.25) under the constraint ̃ (x, t) ≤ min{N, 0≤u

K } ≡ ω(x). |x|γ0

(2.4.26)

By the definition of xt0 , we have u(xt0 , t0 ) = N, and thus − (−Δ) /2 u(xt0 , t0 ) + u2 (xt0 , t0 ) = 𝒜 ∫ α

u(x, t0 ) − u(xt0 , t0 ) |x − xt0 |d+α

dx + N 2 .

(2.4.27)

dr.

(2.4.28)

̃ (x, t0 ) ≤ ω(x) and u ̃ (xt0 , t0 ) = N = ω(xt0 ), we have Since u α

α

−(−Δ) /2 u(xt0 , t0 ) ≤ −(−Δ) /2 ω(xt0 ) ∞

= 𝒜 ∫ (ω(r) − N) ∫ R0

𝕊d

dσ dr |rσ − xt0 |d+α

∞

≤ 𝒜 ∫ (u(r, t0 ) − N)r −d−α ∫ R0

𝕊d

xt | r0

dσ − σ|d+α

43

2.5 Averaged comparison principle |

Remember that ω(r) − N = 0 for r ≤ R0 , and ω(r) − N < 0 for r > R0 . Moreover, as in the case of the function ϕ in (2.4.18), the quantity ∫ 𝕊d

dσ |rσ − xt0 |d+α

is increasing as a function of |xt0 |. Therefore, the maximal value of −(−Δ) /2 ω(xt0 )+N 2 is attained at xt0 = 0. Now, we come back to equation (2.4.27). Using the above estimates and the relation NRγ∗0 = K, we obtain α

𝜕 ̃ (xt0 , t)|t=t0 ≤ −𝒜 u 𝜕t

2

∫ (N − {|x|>R∗ } ∞

= 𝒜σd ∫ ( R∗

K dx K + ( γ0 ) ) |x|γ0 |x|d+α R∗ 2

K r d−1 dr K γ0 − N) d+α + ( γ0 ) r R∗ R∗

2

K K 1 K − α ) + ( γ0 ) 0 α + γ0 Rα+γ αR R∗ ∗ ∗ 1 K 1 0 = α+γ0 (( ). − )𝒜σd + KRα−γ ∗ α + γ0 α R∗ = 𝒜σd (

Since

1 α+γ0

−

1 α

(2.4.29)

0 = K(K/N) /γ0 −1 < 0, we may choose N sufficiently large so that KRα−γ ∗ α

is sufficiently small, so that Theorem 2.4.1.

𝜕 ̃ (xt0 , t)|t=t0 u 𝜕t

< 0 by (2.4.29). This completes the proof of

2.5 Averaged comparison principle We prove in this section a counterpart of Theorem 2.4.1 for radial distributions of solutions used for comparison in the subcritical case α ∈ (1, 2). Again, difficulties related to estimates of fractional Laplacians of radial functions need some special estimates. Theorem 2.5.1 (Averaged comparison principle). Let d ≥ 3 and α ∈ (1, 2) be such that 2α < d. Consider a solution u ∈ 𝒞 2 (ℝd × [0, T]) of system (2.2.1)–(2.2.2) with the radially symmetric initial data u0 ≥ 0, satisfying the integrated bound M(R, 0) =

∫ u0 (x) dx < min{KRd−γ , ϵsRd−α } ≡ bI (R)

for all

R > 0,

(2.5.1)

{|x| 0, ϵ ∈ (0, 1), γ ∈ (0, α), and s = d−α (1.3.6)). Then there exists γ0 = γ0 (α, ϵ) ∈ (0, α) independent of K such that for each initial condition (2.2.3) satisfying condition (2.5.1) with a certain γ ∈ (γ0 , α), the inequality

44 | 2 Large global-in-time solutions to models of chemotaxis 0 ≤ M(R, t) =

∫ u(x, t) dx < bI (R)

(2.5.2)

{|x| 0 and t ∈ [0, T]. First, we need the following asymptotic result in the proof of this comparison principle: Lemma 2.5.2. Let α ∈ (0, 2). The fractional Laplacian of the indicator function of the α unit ball j(r) ≡ (−Δ) /2 1B1 (x) satisfies the relation as r = |x| → 1 𝒪(1) { { j(r) = Cα,d sgn(1 − r)|1 − r|−α + {𝒪(| log |1 − r||) { 1−α {𝒪(|1 − r| )

if if if

0 < α < 1, α = 1, 1 < α < 2,

(2.5.3)

with some number Cα,d > 0. Moreover, j = j(r) is an increasing function on (0, 1) ∪ (1, ∞), j(r) > 0 for r ∈ (0, 1), and j(r) < 0 if r ∈ (1, ∞). Proof. Observe that from definition (23) 1B1 (x) − 1B1 (x − y)

j(r) = j(|x|) = 𝒜 ∫

|y|d+α

{|y| 1 as well as those on the monotonicity of j are clear; it suffices to check the value of the function j(|x|) when 1B1 (x − y) ≠ 0. Now, without loss of generality, we may consider x = (r, 0, . . . , 0) with r > 0, since the problem is rotationally invariant. First, we show that for the half space Π = (−∞, 1] × ℝd−1 , we have α

(−Δ) /2 1Π (x) = Cα,d sgn(1 − r)|1 − r|−α . Indeed, if x ∈ Π, then denoting y = (y1 , y)̄ with ȳ ∈ ℝd−1 , we have α

(−Δ) /2 1Π (x) = 𝒜

∫ {x−y∉Π}

dy |y|d+α

r−1

= 𝒜 ∫ dy1 ∫ −∞ r−1

ℝd−1

dȳ (y12 + |y|̄ 2 )(d+α)/2 ∞

= 𝒜σd−1 ∫ |y1 |−1−α dy1 ∫ −∞

= Cα,d |1 − r|−α .

0

ϱd−2 dϱ (1 + ϱ2 )(d+α)/2 (2.5.4)

2.5 Averaged comparison principle

| 45

Similarly as above, for x ∉ Π, we have α

(−Δ) /2 1Π (x) = −𝒜

∫ {x−y∈Π}

dy = −Cα,d |1 − r|−α |y|d+α

again with ∞

Cα,d = 𝒜σd−1 α−1 ∫ 0

ϱd−2 dϱ . (1 + ϱ2 )(d+α)/2

To complete the proof, it suffices to show that for P = Π \ B1 with the unit ball B1 ⊂ ℝd centered at the origin, the estimate {𝒪(1) α/2 󵄨󵄨 { 󵄨󵄨 󵄨󵄨(−Δ) 1P (x)󵄨󵄨 ≤ {𝒪(| log |1 − r||) { 1−α {𝒪(|1 − r| )

if if if

0 < α < 1, α = 1, 1 0. Proof. A construction of such local-in-time solutions is standard, and it can be based α/2 on the Duhamel formula (see an analog of (2.7.1) below for the semigroup etδΔ e−t(−Δ) ) written for the initial-value problem for system (2.6.1)–(2.6.3). Note, however, that the length of the interval of the existence of the solution constructed in Lemma 2.6.1 depends on ε, δ. The following lemma implies immediately that such a solution can be continued to a common interval [0, T0 ] with T0 > 0, independent of ε > 0 and of δ > 0. Lemma 2.6.2. Let A > 0, s ∈ ℕ, p ∈ ℕ be such that 2p > d/s. Consider a solution u ∈ 𝒞 ([0, T], W s,2p (ℝd )) of the regularized problem (2.6.1)–(2.6.3). i) Then, for all t ∈ [0, T], the inequality T

󵄩2p+1 󵄩 󵄩2p 󵄩󵄩 2p 󵄩󵄩u(t)󵄩󵄩󵄩W s,2p ≤ ‖u0 ‖W s,2p + C ∫󵄩󵄩󵄩u(τ)󵄩󵄩󵄩W s,2p dτ

(2.6.4)

0

holds true with a constant C, independent of ε, δ. In particular, for an initial datum satisfying ‖u0 ‖W s,2p ≤ A, we have the estimate ‖u(t)‖W s,2p ≤ 2A for all t ∈ [0, 1/(4AC)]. ii) Let ‖u‖𝒞([0,T]×ℝd ) be controlled a priori. Then, there exist increasing functions Cs,A (t) < ∞ on [0, ∞) such that inequality t

󵄩2p 󵄩󵄩 󵄩2p 󵄩 2p 󵄩󵄩u(t)󵄩󵄩󵄩W s,2p ≤ ‖u0 ‖W s,2p + ∫ Cs,A (τ)󵄩󵄩󵄩u(τ)󵄩󵄩󵄩W s,2p dτ 0

holds for t ∈ [0, T].

(2.6.5)

2.6 Construction of global-in-time solutions for α ∈ (0, 1)

Proof. Here, we denote the partial derivative 𝜕β = (

𝜕β1 β 𝜕x1 1

,...,

β = (β1 , . . . , βd ) ∈ ℕd , |β| ≤ s. i) Using equation (2.6.1) and the Leibniz rule, we have

𝜕βd β

𝜕xdd

| 51

) for each multiindex

d 2p 2p−1 2p−1 β 𝜕t 𝜕β u dx ≤ ∫(𝜕β u) 𝜕 (ϕε ∗ ∇v) dx ∫(𝜕β u) dx = ∫(𝜕β u) dt 2p−1

≈ ∑ ∫(𝜕β u) j

∇(𝜕j u) ⋅ ∇(𝜕β−j ϕε ∗ v) dx.

(2.6.6)

Here, to obtain second inequality, we have skipped integrals of good sign coming from all the diffusion terms because 2p−1

− ∫(𝜕β u)

(−Δ) /2 (𝜕β u) dx ≤ 0 α

for each p ≥ 1 and α ∈ (0, 2] by the following Stroock–Varopoulos inequality, see [40, Proposition 3.1] as well as [179, Theorem 2.1 and Condition (1.7)] for a proof. Proposition 2.6.3. For α ∈ (0, 2], w ∈ 𝒞c∞ (ℝd ) and q > 1, the following inequality holds true: 4(q − 1) 󵄨󵄨 α/2 q 󵄨2 ∫󵄨󵄨∇ (sgn w |w| /2 )󵄨󵄨󵄨 dx 2 q 4(q − 1) 󵄨󵄨 α/2 q/2 󵄨󵄨2 ≥ ∫󵄨󵄨∇ |w| 󵄨󵄨 dx. q2

∫ sgn w |w|q−1 (−Δ) /2 w dx ≥ α

(2.6.7)

Now, we estimate the terms for j < β and j = β, separately. For j < β, we use the decomposition ϕε ∗ v = K0 ∗ u + K∞ ∗ u with the kernels K0 (x) = ϕ(x)(ϕε ∗ ∇(−Δ)−1 )(x), Evidently, we have K0 ∈ Lq (ℝd ) if q < together with the estimate

K∞ (x) = (1 − ϕ(x))(ϕε ∗ ∇(−Δ)−1 (x)). d . Thus, for u d−1

(2.6.8)

∈ L∞ , we obtain that K0 ∗ u ∈ L∞

󵄩 β 󵄩 󵄩󵄩 β 󵄩 󵄩󵄩K∞ ∗ 𝜕 u󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩𝜕 K∞ 󵄩󵄩󵄩∞ ‖u‖1 for each multiindex β = (β1 , . . . , βd ) with integers βi ≥ 0, i = 1, . . . , d. By an elementary argument, we can also show that K0 ∗ u ∈ L∞ if u ∈ W s,2p (ℝd ) with suitably large s, p. For j = β, we proceed as follows: 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 2p 󵄨󵄨 β 2p−1 ∇(𝜕β u) ⋅ ∇(ϕε ∗ v) dx󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ ∇(𝜕β u) ⋅ ∇(ϕε ∗ v) dx󵄨󵄨󵄨 󵄨󵄨∫(𝜕 u) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2p 󵄨 󵄨 ≤ 󵄨󵄨󵄨∫(𝜕β u) Δ(ϕε ∗ v) dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2p 󵄨 = 󵄨󵄨󵄨∫(𝜕β u) ϕε ∗ u dx󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨

52 | 2 Large global-in-time solutions to models of chemotaxis Using the estimate ‖u‖∞ ≤ C‖u‖W s,n (valid for s > d/n) in the computations above, we obtain the inequality d ‖u‖nW s,n ≤ C‖u‖n+1 W s,n , dt and thus t

󵄩n+1 󵄩 󵄩n 󵄩󵄩 n 󵄩󵄩u(t)󵄩󵄩󵄩W s,n ≤ ‖u0 ‖W s,n + C ∫󵄩󵄩󵄩u(τ)󵄩󵄩󵄩W s,n dτ.

(2.6.9)

0

Now, if sup0≤t≤T ‖u(t)‖W s,n ≤ 2A, then 󵄩n 󵄩 sup 󵄩󵄩󵄩u(t)󵄩󵄩󵄩W s,n ≤ ‖u0 ‖nW s,n + C(2A)n+1 t

0≤t≤T

for all t ∈ [0, T].

Next, choosing T > 0 as the first moment when ‖u(T)‖W s,n = 2A, we obtain the estimate (2A)n ≤ An + CT(2A)n+1 , 1 . which gives immediately that T ≥ 4AC ii) Let us estimate again a generic term in (2.6.6) obtained from the Leibniz formula

󵄨󵄨 󵄨󵄨 2p−1 β−j 󵄨 󵄨 Jβ,j = 󵄨󵄨󵄨∫(𝜕β u) 𝜕 ∇u ⋅ ∇(𝜕j v) dx 󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 Here, the assumption on the radial symmetry of u is crucial because by Lemma 2.1.8, we have the equations 𝜕i 𝜕k v = 𝜕i 𝜕k (−Δ)−1 u = 𝜕i (

xk M(|x|)), |x|d

obtained from equality ∇v(x) ⋅ x = − σ1 |x|2−d M(|x|) in Lemma 2.1.8 with u(x) = 1 1−d 󸀠 R M (R), σd

d

|x| = R (this is identity (2.1.28)), are bounded since u is a priori 𝒞 2 . For s = 1, |j| = 1, we have |𝜕i 𝜕k v| ≤ C(‖u‖∞ + ‖u‖1 ), and consequently Jβ,j ≤

C‖u‖2p (‖u‖∞ + ‖u‖1 ). W s,2p For |j| = 1, s > 1, we recall by formulas (2.6.8) that ∇(𝜕v) = ∇(−Δ)−1 𝜕u = K0 ∗ 𝜕u + K∞ ∗ 𝜕u. By the recurrence assumption 𝜕u ∈ L2p with the norm bounded by C1,A (t), so that ‖K0 ∗ 𝜕v‖L∞ ≤ CC1,A (t). Similarly, we get |K∞ ∗ 𝜕v| = |𝜕K∞ ∗ u| ≤ ‖u‖1 ‖𝜕K∞ ‖∞ . For |j| ≥ 2, s ≥ 2, by recurrence, we infer that ‖∇𝜕j v‖W |j|−1,2p ≤ Cs−1 (t) and 󵄩󵄩 β−j −1 󵄩 󵄩󵄩𝜕 ∇(−Δ) u󵄩󵄩󵄩∞ ≤ C‖u‖W s−1,2p , and we are done. In the following lemma, we pass to the limit ε → 0 and δ → 0 in the regularized problem (2.6.1)–(2.6.3).

2.6 Construction of global-in-time solutions for α ∈ (0, 1)

| 53

Lemma 2.6.4. Let α < 1, n = 2p > d + 1, p ∈ ℕ, and A > 0. For every u0 ∈ L1 (ℝd ) ∩ W 4,n (ℝd ) such that u0 ≥ 0 and ‖u0 ‖W 4,n (ℝd ) ≤ A, there exists a solution u of problem (2.2.1)–(2.2.3). The function u is defined on [0, T0 ], where T0 = 1/(4AC) is defined in Lemma 2.6.2. This solution satisfies u ∈ 𝒳T0 ≡ 𝒞 ([0, T0 ], W 4,n (ℝd )) ∩ 𝒞 1 ([0, T0 ], W 3,n (ℝd )).

(2.6.10)

Moreover, we have sup0≤t≤T0 ‖u(t)‖W 4,n (ℝd ) ≤ 2A. Proof. Let ‖u0 ‖W s,n ≤ A for some A > 0 and n ∈ ℕ. Suppose that a solution uε,δ exists on the interval [0, kTε,δ ], k ∈ ℕ, with uε,δ ∈ 𝒞 ([0, kTε,δ ], W 4,n (ℝd ) ∩ L1 (ℝd )), Tε,δ > 0 being the common existence time for ‖u0 ‖W s,n ∩L1 ≤ 2A with ‖u(., Tε,δ )‖W s,n ∩L1 ≤ 4A. If sup0≤s≤kTε,δ ‖u(., s)‖W 4,n ∩L1 ≤ 2A, then this solution can be continued onto [0, (k +

1 1)Tε,δ ]. By Lemma 2.6.2, we have sup0≤s≤kTε,δ ‖u(., s)‖W 4,n ∩L1 ≤ 2A for kTε,δ ≤ T0 = 2CA so independently of ε > 0, δ > 0. Assume 2p > d + 1. By compactness, we are able to extract a subsequence uεj (., t) ∈ W 3,n ∩ L1 , which is in 𝒞 2 , and the limiting function

solves system (2.2.1)–(2.2.2) with u0 ∈ W 4,n (ℝd ) ∩ L1 (ℝd ).

We also need a technical lemma on a decay property of radial solutions. Lemma 2.6.5. Suppose that u = u(x, t) ≥ 0 is a radial solution of system (2.2.1)–(2.2.2) with u0 satisfying bound (2.4.2) with a sufficiently small ϵ > 0. Moreover, suppose that u satisfies ∫ u(x, t) dx ≤ M, and the estimates 󵄩 󵄩 󵄩 󵄩󵄩 β 󵄩󵄩𝜕 u(t)󵄩󵄩󵄩p + 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p ≤ C(t),

(2.6.11)

with |β| ≤ n, some C(t) and a sufficiently large fixed n. Then limx→∞ |x|α u(x, t) = 0 uniformly on each interval [0, T], T > 0. Proof. The estimate (2.6.11) is, in fact, satisfied for sufficiently smooth solutions, for example, those constructed either in this Section 2.6 or in [178]. Let Ψ ≥ 0 be a smooth bump function supported on an annulus: supp Ψ ⊂ {1 ≤ |x| ≤ 2}, and its scaling ΨR (x) = Ψ( Rx ), R > 0. Define the moment of u by ΛR (t) = ∫ ΨR (x)u(x, t) dx.

(2.6.12)

Computations similar to those in the proof of Theorem 3.8.1 in Section 3.8 lead to the bound 󵄨󵄨 󵄨󵄨󵄨 d 󵄨󵄨 ΛR (t)󵄨󵄨󵄨 ≤ C(M) . 󵄨󵄨 󵄨󵄨 dt Rα 󵄨 󵄨 This, in turn, gives ΛR (t) ≤

C(M, t) , Rα

54 | 2 Large global-in-time solutions to models of chemotaxis which by radial symmetry implies that u(x, t) dx ≤

∫ {|x−x0 |≤1}

C1 , Rα+d−1

(2.6.13)

when |x0 | = R > 0. Indeed, the spherical shell of radius ≍ R and of width ≍ 1 can be covered by ≍ Rd−1 unit balls. On the other hand, ∫

󵄨p 󵄨 (󵄨󵄨󵄨𝜕β u(t)󵄨󵄨󵄨 + u(t)p ) dx ≤ C2

(2.6.14)

{|x−x0 |≤1}

for |β| ≤ n with a sufficiently big n. The condition (2.6.14) implies now that 󵄨 󵄨 󵄨 󵄨 sup (󵄨󵄨󵄨∇u(x, t)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜕xi 𝜕xk u(x, t)󵄨󵄨󵄨) ≤ C3 .

(2.6.15)

{|x−x0 |≤1}

Next, we consider the truncation χ(x − x0 )u(x, t), where χ ≥ 0 has its support in the unit ball. If for some x1 with |x1 − x0 | ≤ 1 max χ(x − x0 )u(x, t) = χ(x1 − x0 )u(x1 , t), x∈ℝd

then, denoting again by u the function χu, from inequalities (2.6.13) and (2.6.15), we obtain 1 󵄨 󵄨󵄨 2 󵄨󵄨u(x, t) − u(x1 , t)󵄨󵄨󵄨 ≤ C|x − x1 | ≤ u(x1 , t). 2 Indeed, if u(x1 , t) ≥ C3 , then 1 u(x , t) ≤ 2 1

u(x, t) dx ≤

∫ {|x−x1 |≤1}

C

Rd−1+α

,

and we are done. Otherwise, if u(x1 , t) < C3 , then d

C1 u(x1 , t) /2+1 ≤

∫ {|x−x1 |≤1}

u(x, t) dx ≤

C . Rd−1+α

In both the cases, for α < 1, we get the conclusion since the inequality

satisfied for α ≤

. 2 d−1 d

d−1+α d/2+1

≥ α is

Proof of Theorem 2.3.1. The proof of this theorem is a standard application of Lemma 2.6.4, Lemma 2.6.2, and the pointwise comparison principle in Theorem 2.4.1. Let us fix α < 1, n = 2p ≥ d + 1, p ∈ ℕ, ε < 1. By Lemma 2.6.4, there exists T0 > 0 such that the system (2.2.1)–(2.2.2) has a solution u ∈ 𝒳T0 , with the space 𝒳T0 defined

2.7 Construction of the unique global-in-time solutions for α ∈ (1, 2)

| 55

in (2.6.10). We will show that this solution can be continued onto the interval [0, T1 ], to u ∈ 𝒳T1 , with T1 − T0 ≥ Δ(A, [T0 ], ε) > 0. First, observe that by assumptions of Theorem 2.3.1 and property (2.6.10), there exist K, N, and γ < α such that 0 ≤ u0 (x) < ϵs min{N, |x|Kγ , |x| α }, so that by Lemma 2.6.5, condition (2.4.1) of Theorem 2.4.1 is satisfied. Consequently, the estimate u(x, t) < min{N,

K ϵs , } |x|γ |x|α

holds for each t ∈ [0, T0 ]. In particular, by Lemma 2.6.2, we infer that ‖u(t)‖W 4,n ≤ H(A, [T0 ] + 2, ε). Take T 󸀠 < T0 , close to T0 . By Lemma 2.6.4, the solution v with the initial condition v(0) = u(T 󸀠 ) exists on an interval of length (at least) Δ = Δ(A, [T0 ], ε). Therefore, the solution of the original Cauchy problem can be continued onto [0, T0 + Δ/2], which shows the claim.

2.7 Construction of the unique global-in-time solutions for α ∈ (1, 2) In this Section, we prove Theorem 2.3.3 by constructing global-in-time solutions in the homogeneous Morrey space M p (ℝd ). Let us begin with auxiliary estimates, which are somewhat hidden in [2, Lemma 3.1], so we formulate this explicitly and prove in the following Proposition 2.7.1 (Comparison of the concentration and the Morrey norm). There exists a constant c(d) ∈ (0, 1) such that for each nonnegative and radially symmetric d v ∈ M /κ (ℝd ) with κ ∈ [1, d), we have the inequality c(d)||v||M d/κ ≤ sup Rκ−d ∫ v(y) dy ≤ |v||M d/κ . R>0

{|y| 0 is a number depending on the dimension d only. Thus, by the radial symmetry of v, we obtain d−1

∫ {|x−x0 | 1, this solution is sufficiently regular (for example, u ∈ 𝒞 2 (ℝd × [0, T))), which can be proved repeating the reasoning from [114]. This solution is radial and nonnegative if the corresponding initial datum is so, by a usual comparison argument. To prove that this local-in-time solution can be extended to all t > 0, it suffices to show that neither |u(t)||M d/α nor ‖u(t)‖∞ can blowup in a finite time. By assumptions Theorem 2.3.3 and by Remark 2.3.4, there exist constants K > 0 and γ ∈ (0, α) such that ∫ u0 (x) dx < min{KRd−γ , ϵ {|x| 0. d−α

Then, applying Proposition 2.7.1, one can immediately check that u0 ∈ M p (ℝd ) with p = d/γ > d/α. Thus, by the comparison principle proved in Theorem 2.5.1 combined

58 | 2 Large global-in-time solutions to models of chemotaxis with Proposition 2.7.1, there exists a number C independent of T such that |u(t)||M d/α ≤ C and |u(t)||M p ≤ C for all t ∈ [0, T]. Next, we estimate the L∞ norm of both sides of equation (2.7.1) using inequalities (42) and (2.7.5) with 1r = p1 − d1 in the following way: t

󵄩 󵄩 󵄩 󵄩󵄩 󵄩 −1− d 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩Tα (t)u0 󵄩󵄩󵄩∞ + C ∫(t − s) α αr 󵄩󵄩󵄩u(s)󵄩󵄩󵄩∞|∇Ed ∗ u(s)||M r ds t

0

≤ ‖u0 ‖∞ + C ∫(t − s)

− α1 − dα ( p1 − d1 ) 󵄩 󵄩

󵄩 󵄩󵄩u(s)󵄩󵄩󵄩∞|u(s)||M p ds.

0

Thus, the L∞ norm of the solution is controlled locally in time, thanks to a singular Gronwall-type argument (see [104, Lemma 1.2.9] or [134, 1.2.1, 7.1.1]), because −

d 1 d 1 1 − ( − ) ∈ (−1, 0) for p > , α α p d α

and because sups>0 |u(s)||M p < ∞ by Theorem 2.5.1 combined with Proposition 2.7.1.

2.8 Large global-in-time solutions of the parabolic-parabolic Keller–Segel system on the plane Although the doubly parabolic Keller–Segel system is not our main topic in this book, we show in this section that this case is different than the parabolic-elliptic. Namely, the critical value of mass depends on the diffusivity coefficient in the equation for the evolution of chemoattractant (12), and each mass may lead to a global-in-timesolution, even if the initial data is a finite signed measure, if this coefficient is large enough. The presentation is based on the paper [37]. In general, these solutions need not be unique, even if we limit ourselves to nonnegative solutions, which has been remarked in the Introduction in discussion of selfsimilar solutions (13) studied in [3]. Thus, we consider in this section the doubly parabolic version of the Keller–Segel model of chemotaxis with consumption term ut = ∇ ⋅ (∇u − u∇v),

εvt = Δv − γv + u,

x ∈ ℝ2 , t > 0,

(2.8.1)

2

x ∈ ℝ , t > 0.

(2.8.2)

v(⋅, 0) = 0,

(2.8.3)

We suppose that the initial condition u(⋅, 0) = u0 ,

is a finite Radon measure u0 ∈ ℳ(ℝ2 ). We choose v(x, 0) = 0 for simplicity, however, the analogous computations could be done with every sufficiently regular v(x, 0) with

2.8 Parabolic-parabolic Keller–Segel system | 59

∇v0 ∈ L2 (ℝ2 ). Here, the constant parameter ε > 0 is related to the diffusion rate of the chemical, and usually in applications ε is small since the chemoattractant diffuses much faster than the population. We are interested, however, in arbitrary positive values of ε. The coefficient γ ≥ 0 is the consumption/degradation rate of the chemical. In that interpretation u(x, t) ≥ 0 and v(x, t) ≥ 0, which is the consequence of the nonnegativity of the initial data u0 (x) ≥ 0, v(x, 0) ≥ 0, see [18], [86] and [172]. However, for the existence of solutions, we do not need this assumption, so we prefer to deal with arbitrary (sign changing) solutions as was in [18] or [172]. It is well known that the integral ∫ℝ2 u(x, t) dx (and thus, total mass M = ∫ℝ2 u(x, t) dx for nonnegative solutions) is conserved for sufficiently regular solutions of (2.8.1)–(2.8.3). We will show this property in Remark 2.8.5 after the proof of the main theorem. The limit case ε = 0 is called the parabolic-elliptic Keller–Segel model, and has been much more studied than the doubly parabolic one with ε > 0. For the relations between those two systems as ε ↘ 0, see [214, 11, 172]. Recall that in the parabolic-elliptic case (ε = 0), the critical value of mass is M = 8π, and measures as initial conditions with atoms bigger than 8π are obstructions, even for the local-in-time existence of solutions. Many interesting results on the doubly parabolic Keller–Segel are presented in [106]. In the following theorem, we show that one should not expect any universal value of critical mass determining the existence of solutions to parabolic-parabolic Keller– Segel model (2.8.1)–(2.8.3) with sufficiently large ε > 0. Here, as usual, we deal with mild solutions, solutions of the Duhamel formulation of problem (2.8.1)–(2.8.3), see equations (2.8.6)–(2.8.8). Theorem 2.8.1. For each u0 ∈ ℳ(ℝ2 ), there exists ε(u0 ) > 0 such that for all ε ≥ ε(u0 ) the Cauchy problem (2.8.1)–(2.8.3) has a global-in-time mild solution satisfying u ∈ 𝒞w ([0, ∞); ℳ(ℝ2 )) and v ∈ 𝒞 ([0, ∞); L1 (ℝ2 )). This is a classical solution of the system (2.8.1)–(2.8.2) for t > 0, and satisfies 󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p < ∞ t>0

(2.8.4)

for each p ∈ [1, ∞]. Theorem 2.8.1 improves results for the parabolic-parabolic Keller–Segel model (ε > 0) in [18], [86] and [186], where the global existence of solutions for M < 8π on the whole plane has been considered. Note that there exist large (unstable) stationary solutions in balls, see [18, Ch. 6]. All those results show that, unlike the parabolic-elliptic case, there is no universal threshold value of mass for the local existence of solutions as well as for the globalin-time existence.

60 | 2 Large global-in-time solutions to models of chemotaxis Remark 2.8.2. In general, solutions of the Cauchy problem (2.8.1)–(2.8.3) need not be unique. This striking property is seen when we consider certain radially symmetric nonnegative selfsimilar solutions for the system (2.8.1)–(2.8.2) with γ = 0—, which are of the scaling invariant form (13) with some functions U, V of one variable decreasing exponentially to 0 as |x| → ∞. They have been constructed using ODE methods in [3], and they correspond to the initial data in (2.8.1)–(2.8.2) 1 |x| u0 = lim U( ) = Mδ0 , t→0 t √t

with the Dirac measure δ0 , attained in the sense of weak convergence of measures, and . v0 = lim V( ) = 0 t→0 √t a. e. and in L1 (ℝ2 ). In particular, for 0 < ε ≤ 1/2, they exist exactly in the range M ∈ [0, 8π). However, for ε > ε∗ (≈ 0.64) there exist selfsimilar solutions with M ∈ [0, Mε ] with at least two solutions for each M ∈ (8π, Mε ), Mε > 8π, and even limε→∞ Mε = ∞ ε−1 since it follows from [3, Theorem 4] that Mε ≥ 4π . This is, to the best of our knowle log ε edge, the first nontrivial example of nonuniqueness of mild solutions to a chemotaxis system with measures as initial conditions. For further results on selfsimilar solutions (for example, uniqueness in some cases), see [106, Section 3]. Then, the authors of [106] proved that there exist mild solutions of any mass, provided that ε > 0 is sufficiently large. Finally, let us formulate an important consequence of our result. For arbitrary M > 0 and for all ε ≥ ε(M), Theorem 2.8.1 provides us with a mild solution of the Cauchy problem (2.8.1)–(2.8.3) with u0 = Mδ0 and γ = 0. Note that if M > 0 is sufficiently small, this is a unique solution in the space 𝒳 , see Remark 2.8.6 below. Moreover, by a standard argument (see [197]), one may show that each such solution has the selfsimilar form (13) and, moreover, it is a radial function. A perusal of the proof of [3, Theorem 4] shows that such solutions are also mild solutions of problem (2.8.1)– (2.8.3) and satisfy properties stated in Theorem 2.8.1. By Remark 2.8.2, there exists another selfsimilar (and radial) solution with the same value of M > 8π. One of them can be constructed by the method applied in the proof of Theorem 2.8.1, the others being unstable with respect to the iteration method used for solving (2.8.6). We will use Lq − Lp estimates for the heat semigroup on the plane (39), (40) and an immediate consequence of inequality (39) 󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩etΔ u0 󵄩󵄩󵄩p ≤ C‖u0 ‖1 , t>0

together with the inequality 󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩etΔ μ󵄩󵄩󵄩p ≤ C‖μ‖ℳ(ℝ2 ) , t>0

which holds true for all μ ∈ ℳ(ℝ2 ).

(2.8.5)

2.8 Parabolic-parabolic Keller–Segel system

| 61

The mild formulation of system (2.8.1)–(2.8.2), together with initial conditions (2.8.3) is the integral equation (also known as the Duhamel formula) u(t) = etΔ u0 + ℬ(u, u)(t),

(2.8.6)

where the quadratic form ℬ is defined by t

ℬ(u, z)(t) = − ∫(∇e

(t−s)Δ

) ⋅ (u(s) Lz(s)) ds,

(2.8.7)

0

with the solution operator of (2.8.2) t

Lz(t) = ε−1 ∫(∇eε

−1

(t−s)(Δ−γ)

)z(s) ds.

(2.8.8)

0

The existence of solutions of the quadratic equation (2.8.6) is established by the usual approach using the Banach contraction argument in a suitable functional space of vector-valued functions. In our case, that space is denoted by p

2

ℰp = {u ∈ Lloc ((0, ∞); L (ℝ )) : sup t ∞

t>0

1−1/p 󵄩 󵄩

󵄩 󵄩󵄩u(t)󵄩󵄩󵄩p < ∞},

and the norm || ⋅ ||p in ℰp is defined as 󵄩 󵄩 ||u|| p ≡ sup t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p < ∞. t>0

(2.8.9)

In this section, only the || ⋅ ||p norm is not related to the radial concentrations (20), (19). Then, we will show that actually u ∈ 𝒳 ≡ 𝒞w ([0, ∞); ℳ(ℝ2 )) ∩ ℰp , which means, moreover, that the mapping [0, ∞) ∋ t 󳨃→ ∫ℝ2 u(x, t)φ(x) dx is continuous for each test function φ ∈ 𝒞0 (ℝ2 ). Remark 2.8.3. Note that solutions of the equation u = y0 + ℬ(u, u) (more general than (2.8.6)) in a Banach space (𝒴 , ‖ ⋅ ‖𝒴 ) provided by that contraction argument (or, equivalently, by the Picard iteration scheme) are locally unique, but they need not be unique in general as Figure 2.1 shows for 𝒴 = ℝ, and for the quadratic equation u = y0 + ηu2 1 with a fixed η > 0 and |y0 | < 4η . The proof of Theorem 2.8.1 is split into two parts. In the first, solutions of (2.8.6) are constructed in ℰp with a fixed p ∈ (4/3, 2). Then, they are shown to attain the initial data in the weak sense, that is, they belong to 𝒳 = 𝒞w ([0, ∞); ℳ(ℝ2 )) ∩ ℰp . The first part of the proof is based on two auxiliary facts: Lemma 1.1.1 and

62 | 2 Large global-in-time solutions to models of chemotaxis

Figure 2.1: Two solutions u1 and u2 of the quadratic equation u = y0 + ηu2 .

Lemma 2.8.4. Let p ∈ (4/3, 2). The bilinear form ℬ is bounded from ℰp × ℰp into ℰp : ||ℬ(u, z)|| p ≤ η|| u|| p ||z|| p with a constant η = η(ε) independent of u, z, and γ, such that η(ε) → 0 as ε → ∞. Proof. First, we estimate the Lq -norm of the linear operator L defined in (2.8.8) acting on z ∈ Lp (ℝ2 ) using the estimates (39)–(40). Assuming that 1 ≤ p ≤ q ≤ ∞, p < ∞, and p1 − q1 < 21 , we obtain t

−1/2+1/q−1/p −γε−1 (t−s) 󵄩 󵄩 󵄩󵄩 −1 −1 󵄩󵄩z(s)󵄩󵄩󵄩 ds e 󵄩󵄩Lz(t)󵄩󵄩󵄩q ≤ Cε ∫(ε (t − s)) 󵄩p 󵄩 0

t

1 1 󵄩 󵄩 ≤ Cε− /2−1/q+1/p ∫(t − s)− /2+1/q−1/p s1/p−1 ( sup s1−1/p 󵄩󵄩󵄩z(s)󵄩󵄩󵄩p ) ds

0 −1. Next, we may prove the estimate of the bilinear form ℬ. In the following computations, we fix the exponents p and q to have p 2p 4 < p ≤ 2 ≤ p󸀠 = −1 2 p r 2 q

1 p

+

1 q

and

(2.8.11)

< 1 defines the exponent r ∈ (1, p). 1 1 3 + − > −1, q p 2

2.8 Parabolic-parabolic Keller–Segel system

| 63

as well as p1 − q1 < 21 . Thus, using the inequality (2.8.10), we have the following estimate of the bilinear form ℬ: t

󵄩 󵄩󵄩 −1/2+1/p−1/r 󵄩 󵄩󵄩u(s) Lz(s)󵄩󵄩󵄩 ds 󵄩󵄩ℬ(u, z)(t)󵄩󵄩󵄩p ≤ C ∫(t − s) 󵄩r 󵄩 0

t

1 󵄩 󵄩 󵄩 󵄩 ≤ C ∫(t − s)− /2+1/p−1/r 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p 󵄩󵄩󵄩Lz(s)󵄩󵄩󵄩q ds

0

≤ Cε

−1/2−1/q+1/p

t

1 󵄩 󵄩 ∫(t − s)− /2−1/q s1/p−1 ( sup s1−1/p 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p )

0 −1 and p1 − 1 + 2σ in the proof. The above inequalities lead to

1 q

−

1 2

= −1 +

1 r

−

1 2

> −1 as requested

||u|| σ ≤ ||etΔ u0 ||σ + ||B(u, u)|| σ ≤ C‖u0 ‖ℳ(ℝ2 ) + C|| u|| 2p . The proof in the case p = ∞ is completely analogous. Now, we proceed to the proof that this solution attains the initial data in the weak sense; this satisfies u ∈ 𝒞w ([0, ∞); ℳ(ℝ2 )). For that purpose, first we define for a fixed p ∈ (4/3, 2) a subspace of ℰp 𝒴 = L∞ ((0, ∞); ℳ(ℝ2 )) ∩ ℰp endowed with the norm ‖u‖𝒴 = ess supt>0 ‖u(t)‖ℳ(ℝ2 ) +|| u|| p . We will show that Lemma 1.1.1 applies to equation (2.8.6) in the space 𝒴 , that is, the following estimate ‖ℬ(u, v)‖𝒴 ≤ η(ε)‖u‖𝒴 ‖v‖𝒴 is valid with η(ε) → 0 as ε → ∞. The L1 estimate (2.8.13) together with the bilinear bound for ℬ in the norm of ℰp established in Lemma 2.8.4, and the estimate ‖etΔ u0 ‖1 ≤ ‖u0 ‖1 , show that for fixed u0 ∈ ℳ(ℝ2 ) and ε sufficiently large, equation (2.8.6) has a solution in 𝒴 . Next, to prove that the constructed solution is in fact in the space 𝒳 = 𝒞w ([0, ∞); ℳ(ℝ2 )) ∩ ℰp ⊂ 𝒴 , we will show that u(t) ⇀ u(s) as t ↘ s ≥ 0 in the sense of the weak convergence of measures. For that purpose, we need only to show that ℬ(u, u)(t) ⇀ ℬ(u, u)(s) as t ↘ s; in particular ℬ(u, u)(t) ⇀ 0 as t → 0. Indeed, the sufficiency of that property is seen from the representation u(t) − u(s) = etΔ u0 − esΔ u0 + ℬ(u, u)(t) − ℬ(u, u)(s),

2.8 Parabolic-parabolic Keller–Segel system

| 65

and from the fact that the heat semigroup is weakly continuous in ℳ(ℝ2 ): etΔ u0 ⇀ esΔ u0 as t ↘ s ≥ 0. Now we write ℬ(u, u)(t) − ℬ(u, u(s)) = I1 + I2 ,

where

s

I1 = ∫ ∇(e(s−σ)Δ − e(t−σ)Δ ) ⋅ (u(σ)Lu(σ)) dσ 0

and t

I2 = − ∫ ∇e(t−σ)Δ ⋅ (u(σ)Lu(σ)) dσ. s

Putting u = z in (2.8.13), we get 󵄩 󵄩󵄩 −1/2 󵄩󵄩u(σ)Lu(σ)󵄩󵄩󵄩1 ≤ Cσ .

(2.8.14)

Indeed, this is a consequence of 1 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 1/p−1 1/p󸀠 −1/2 σ ||u|| p ≤ Cσ − /2 ||u|| 2p . 󵄩󵄩u(σ)Lu(σ)󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩u(σ)󵄩󵄩󵄩p 󵄩󵄩󵄩Lu(σ)󵄩󵄩󵄩p󸀠 ≤ Cσ

Since the two dimensional Gauss–Weierstrass kernel (38) satisfies x |x|2 𝜕 |x|2 3 ∇G(x, t) = t − /2 (2 − )(8πt)−1 1/2 exp(− ), 𝜕t t 4t t we have 󵄩 󵄩󵄩 −1/2 󵄩󵄩∇(G(⋅, s − σ) − G(⋅, t − σ))󵄩󵄩󵄩1 ≤ C(t, s)(s − σ) with C(t, s) → 0 as t ↘ s ≥ 0 since (t − σ) − (s − σ) = (t − s) ≤ (t − σ). By the Lebesguedominated convergence theorem and t

1

1

∫(t − σ)− /2 σ − /2 dσ = π,

(2.8.15)

0

we arrive at ‖I1 ‖1 → 0. t Concerning I2 , we note that if s > 0, then by estimates (2.8.14), (40), and ∫s (t − 1

1

1

1

1

σ)− /2 σ − /2 dσ = ∫s/t (1 − ρ)− /2 ρ− /2 dρ → 0 as t ↘ s, we get ‖I2 ‖1 → 0 as t ↘ s > 0. For s = 0,

taking a smooth compactly supported test function φ ∈ 𝒞01 (ℝ2 ), we see that t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨− ∫ ∫ ∇ ⋅ (e(t−s)Δ (u(s)Lu(s))) ds φ(x) dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 ℝ 0

66 | 2 Large global-in-time solutions to models of chemotaxis 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 (t−s)Δ 󵄨 = 󵄨󵄨∫ ∫ e (u(s)Lu(s)) ⋅ ∇φ(x) dx ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 ℝ2 󵄨 t

󵄩 󵄩 ≤ C ∫󵄩󵄩󵄩u(s)Lu(s)󵄩󵄩󵄩1 ‖∇φ‖∞ ds 0

t

1

≤ C ∫ σ − /2 dσ 0

‖∇φ‖∞ → 0

as t → 0. Since such test functions are dense in 𝒞0 (ℝ2 ), and ‖ℬ(u, u)(t)‖1 are uniformly bounded for t > 0, the weak convergence ℬ(u, u)(t) ⇀ 0 as t → 0 follows, see an analogous reasoning in [131, Theorem 4.1]. Finally, we define v = Lu, where the linear operator L is given by formula (2.8.8). Using estimate (2.8.5) with p = 1, it is an elementary calculation to show that v ∈ 𝒞 ([0, ∞); L1 (ℝ2 )), and it corresponds to the zero initial condition in the following sense: limt→0 ‖v(t)‖1 = 0. Remark 2.8.5. It is easy to see that the integral ∫ℝ2 u(x, t) dx = ∫ℝ2 u0 (x) dx is conserved during the evolution of mild solutions satisfying (2.8.6). Indeed, ∫ℝ2 etΔ u0 (x) dx = ∫ℝ2 u0 (x) dx and ∫ℝ2 ℬ(u, u)(s) dx = 0 for each s ∈ (0, t), the latter being the consequence of the full divergence form of ℬ(u, u)(s) ∈ L1 (ℝ2 ). This is seen by applying estimate (2.8.14) together with (40) and (2.8.15) to the nonlinear term in (2.8.6). Remark 2.8.6. For a fixed ε and for sufficiently small ‖u0 ‖ℳ(ℝ2 ) , the solution of (2.8.1)– (2.8.3) constructed in the space 𝒳 is unique. Indeed, the existence of the unique local-in-time solution is proved in the space 𝒳T = 𝒞w ([0, T]; ℳ(ℝ2 )) ∩ {u : (0, T) → Lp (ℝ2 ) : sup0 0, the right-hand side of equation (2.8.6) defines the contraction on a suitably chosen ball {u : sup0 0, Z > 0}, as well as the sector 𝒞 = {(X, Z) : X > 0, Z > 0, X > (d − 2)Z}, are invariant for system (2.9.3)–(2.9.4) so that each solution of equation (2.9.1) with M(0) ≥ 0, Mr (0) ≥ 0 is positive and nondecreasing. We are interested in the, so called, eternal solutions of this dynamical system defined on the whole real line ℝ ∋ τ. The linearization of system (2.9.3)–(2.9.4) at the point (2(d − 2), 2) has two complex conjugate eigenvalues with negative real part if 3 ≤ d ≤ 9, and two negative real eigenvalues if d ≥ 11. Hence, there exists a separatrix joining the unstable point (0, 0) (when τ → −∞) with the stable point (2(d − 2), 2) (when τ → ∞). Its slope at the origin is equal to 1/d. Observe that if 3 ≤ d ≤ 9, then r 2−d M(r) is not monotone, since the separatrix turns around the point (2(d − 2), 2) infinitely many times. For d = 2 and d ≥ 10, there is a unique scroll of such a curve, since the eigenvalues of the linearization of the vector field at (2(d − 2), 2) are real. It can be shown that the separatrix joining the stationary points (0, 0) and (2(d − 2), 2) on Figure 2.2 is the unique trajectory with X(τ) > 0 for all τ ∈ ℝ, and this satisfies X(−∞) = 0, Z(−∞) = 0. Therefore, all the stationary solutions are parameterized by, say, their values Z(τ) at τ = 0 corresponding to σ1 M(r) at r = 1. This determines the d

68 | 2 Large global-in-time solutions to models of chemotaxis

Figure 2.2: Stationary solutions.

number of such stationary solutions with a given value M(1). For 3 ≤ d ≤ 9, there might be multiple stationary solutions with the same value of M(1). All of them satisfy the relation limτ→∞ Z(τ) = 2, so that relation (2.9.2) follows. This leads to the following result (see [18, Section 6]) for 3 ≤ d ≤ 9: Proposition 2.9.1. There exist radial stationary solutions with their radial concentrations (that their σd Z functions) oscillating in r (and, therefore, their densities not monotone along r) infinitely many times around the value 2σd .

2.9.2 Selfsimilar solutions of the classical Keller–Segel system Here we gather some facts on the (forward) selfsimilar solutions, whose proofs can be found in [16, 17]. For further development, involving another method, see [47]. Solutions invariant under the scaling leaving system (2.1.1)–(2.1.2) unchanged are of the form u(x, t) = 1t U( √xt ), and their initial data are homogeneous of degree −2. Thus, radial positive selfsimilar solutions, which are not stationary have as their initial values u0 = ϵuC with small ϵ > 0, anyway ϵ < 1. 2 d For them, the integrated density is of the form M(r, t) = σd t /2−1 ζ ( rt ), and the profile ζ = ζ (y), y =

r2 󸀠 , t

=

d , dy

satisfies the ordinary differential equation

1 d−2 󸀠 d−2 1 ζ 󸀠󸀠 + ζ 󸀠 − ζ − ζ + d/2 ζζ 󸀠 = 0, 4 2y 8y 2y

ζ (0) = 0,

d

ζ (y) ≈ 2ϵy /2−1 , y → ∞. (2.9.6)

It can be shown that each solution of problem (2.9.6) verifies estimates ζ (y) ≤ (1 −

2 d/2 d )y + 4(d − 1)y /2−1 , d

lim y1− /2 ζ (y) ∈ (0, ∞). d

y→∞

(2.9.7) (2.9.8)

2.9 Comments on special solutions |

69

Figure 2.3: Selfsimilar solutions.

The change of variables τ =

1 2

log y, ̇ =

d , dτ

X(τ) = 2y2− /2 ζ 󸀠 (y), d

Z(τ) = y1− /2 ζ (y), d

similar to that for stationary solutions, leads to a nonautonomous system in the plane (X, Z) e2τ ((d − 2)Z − X), Ẋ = (2 − Z)X + 2 Ż = X − (d − 2)Z.

(2.9.9) (2.9.10)

Selfsimilar solutions emanate from the origin (0, 0) and terminate at the points (2(d − ̇ 2)ϵ, 2ϵ), ϵ ≪ 1, moving in the sector {X > (d − 2)Z} since Z(τ) > 0 for τ ∈ ℝ corresponḋ ∈ (d − 2, d). ing to r ∈ (0, ∞), with the initial slope limτ→−∞ X(τ) ̇ Z(τ) Two curves on the right-hand side on Figure 2.3 above correspond to stationary solutions, the other to self-similar solutions. We show that for each point on the line ℓ = {X = (d − 2)Z}, between (0, 0) and (2(d − 2), 2), is a termination point of a curve describing selfsimilar solution. In other words we show Proposition 2.9.2. For each c ∈ (0, 2(d − 2)), there is a unique selfsimilar solution with the initial condition |x|c 2 . In other words, for each ε ∈ (0, 1), there exists a selfsimilar solution emanating from εuC . For another, completely different argument proving this Proposition 2.9.2, see [47]. To get this conclusion, we will compare trajectories of that nonautonomous system with the autonomous system (2.9.3)–(2.9.4) for stationary solutions of equation (2.9.1). Let (X, Z) be the solution describing a selfsimilar solution, (Xs , Zs )—a solution for steady states, and (XN , ZN )—a solution of the auxiliary autonomous dynamical system Ẋ = (2 − Z)X + N((d − 2)Z − X), (2.9.11) Ż = X − (d − 2)Z,

(2.9.12)

70 | 2 Large global-in-time solutions to models of chemotaxis called here the N-system, which will be used for comparison up to τN , satisfying exp(2τN )/2 ≤ N. Clearly, for N = 0, this is the system (2.9.3)–(2.9.4). For large N ≥ 0, the stationary point (2(d − 2), 2) of the N-system is stable. Indeed, the eigenvalues of the linearization at this point 1 (−(N + d − 2)2 ± √(N + d − 2)2 − 8(d − 2)) 2 are negative for large N, whereas for the original system (2.9.3)–(2.9.4), they have negative real parts. For a fixed N (large) and each τ ∈ (−∞, τN ), we have XN (τ) ≤ X(τ) ≤ Xs (τ). Since XN (τ) converges to 2(d − 2) as τ → +∞ by the stability property of N-system, X(τ) does. Then, we look for the asymptotics of each solution (X, Z) with different τ0 , and X(τ0 ) 1 X(τ0 ), that is, passing through the vertical line cutting the fixed, Zs (τ0 ) ≤ Z(τ0 ) ≤ d−2 sector 𝒞 at some point X = X(τ0 ). They all go to a point (Z/(d−2), Z) with X(τ0 )/(d−2) < Z < 2 by the argument concerning the quantity Y = X − (d − 2)Z, by comparison with the N-system with suitably large N, see below. So each 0 < Z < 2 is a terminal point on the line ℓ of a trajectory describing selfsimilar solution. Since y = exp(2τ)/2 = r 2 /t, this corresponds to any initial condition |x|c 2 , c ∈ (0, 2(d − 2)), between 0 and uC , |x| = r, leading to a selfsimilar solution. Let us come back to the asymptotics of Y(τ) as τ → +∞. First, Y = X − (d − 2)Z ≥ 0 in the sector 𝒞 . Computing the τ-derivative of Y, we obtain e2τ Y. Ẏ = (2 − Z)X − (d − 2)Y − 2 This implies e2τ ̇ Y(τ) + (d − 2)Y(τ) + Y(τ) ≤ C = const 2 since (2 − Z)X ≥ 0 in 𝒞 and Y(τ0 ) ≥ 0. As a consequence, we get τ

Y(τ) ≤ e

−(d−2)τ

e2s e2τ exp(− ) ∫ e(d−2)s exp( ) ds. 2 2 0

Since the limit of the right-hand side as τ → +∞ is equal to 0, we arrive at the conclusion. To see the latter, let z = e2τ so that τ = 21 log z. The integral behaves asymptotically like z0

∫ z k ez 0

by the de l’Hospital rule, where k = − d−2 2

Y is z0

dz ≈ z0k−1 ez0 z

d−2 , and the factor in front of the upper bound for 2

e−z0 , which shows the claim above.

2.9 Comments on special solutions | 71

Note that backward selfsimilar solutions (selfsimilar blowing up solutions) are discussed in the radial case in [220], see [136]. The role of selfsimilar solutions in the evolution of general solutions for the classical Keller–Segel system in two dimensions is discussed in [197, 200, 89, 118]. In particular, the intermediate asymptotics is shown to be selfsimilar. For results in this spirit for other mean field models, see [30, 207, 206]. Note that, in higher dimensions, the asymptotics for the Nernst–Planck–Debye–Hückel system for the evolution of electrically charged particles is diffusion-dominated, see [30].

3 Blowups Recently, some new results on the blowup of solutions to the Keller–Segel system with classical diffusion, i. e. to problem (3.2.1)–(3.2.3), appeared in [195, 163, 164, 50, 51, 28, 52] with a new strategy of the proofs involving local momenta of (most frequently) radial solutions, and with improved sufficient conditions in terms of the initial datum u0 . As we have remarked already, it is also well known that the Keller–Segel system with fractional diffusion (2.2.1)–(2.2.2)—that is (3.8.1)–(3.8.2) in this Chapter— possesses local-in-time solutions which cannot be continued to the global-in-time ones, see [39, 194, 41] for some relevant results. Criteria for a blowup of solutions with large concentrations can be expressed in terms of related critical Morrey space norms (see Remark 3.8.2 below for more detailed presentation), and we have found that the size of such a norm is also critical for the global-in-time existence versus finite time blowup. Such results for radially symmetric (and N-symmetric) solutions of the d-dimensional classical Keller–Segel model with d ≥ 2 and α = 2 have been recently studied in [50, 51]. We present in this and the following Chapters 3, 4 two novel approaches to blowup phenomena for chemotaxis systems. The first uses local moments (going back to ideas in [194, 163]) defined by bump functions like ψ(x) = (1−|x|2 )2+ and their rescalings. This has been begun in [50], used in [64, 51, 28] and refined in [52]. The second is based on classical approach in the seminal paper [125] by H. Fujita where large values of a solution u are detected by testing radially symmetric solutions by an approximative identity, in fact, the solution of the backward heat equation that concentrates at the origin at time t = T is used as a weight. This permits us to improve the sufficient conditions for the blowup expressed in terms of a functional norm of u0 as in [65]. But first we go back to the primary approach to blowup begun in [148, 39, 15, 194], and present results in [41] obtained with the use of classical virial computations for moments with power weights, somewhat more sophisticated than those mentioned in references above. Here, it is worth noting that the idea of using the second moment of solutions with the weight function |x|2 is reminiscent of the kinetic theory where driftdiffusion systems have their kinetic predecessors derived from the microscale level description of systems interacting particles.

3.1 Solutions blowing up in a finite time It is well known that if d = 2, the condition leading to a finite time blowup, i. e. lim sup u(x, t) = ∞

t↗T, x∈ℝd

for some

0 < T < ∞,

is expressed in terms of a single parameter, viz. mass, that is M > 8π, see e. g. [15, 50, 51]. https://doi.org/10.1515/9783110599534-003

74 | 3 Blowups In all these cases, at the blowup time 0 < T < ∞, we have limt↗T ‖u(x, t)‖∞ = ∞ ([15, 41]). In order to consider solutions which are unbounded for large x, it is of interest to adopt the following definition of the blowup: there exists T > 0 such that for some R>0 lim sup sup u(x, t) = ∞, t↗T

|x| 0 looks like the Chandrasekhar steady state singular solution uC , namely, it satisfies C1 |x|−2 ≤ u(x, T) ≤ C2 |x|−2 with convergence in L1 as t → T. We will discuss blowup phenomena also in next chapters. If d ≥ 3, a sufficient condition for blowup for an initial condition (not necessarily radial) is that u0 is highly concentrated, namely (

∫ℝd |x|γ u0 (x) dx ∫ℝd u0 (x) dx

d−2 γ

)

≤ c̃d,γ M,

(3.1.1)

for M = ∫ℝd u0 (x) dx, some 0 < γ ≤ 2 and a (small, explicit) constant c̃d,γ > 0, see [41, (2.4)] and a more general Lemma 3.1.1 below. Since |u0|M d/2 ≥ C̃ d,γ M(

M ) ∫ℝd |x|γ u0 (x) dx

d−2 γ

for some constant C̃ d,γ > 0 and all u0 ∈ M /2 ∩ L1 , see [41, (2.6)], this means that, when d the sufficient blowup condition (3.1.1) is satisfied, the Morrey space M /2 norm of u0 satisfying condition (3.1.1) must be (very!) large: d

|u0|M d/2 ≥

C̃ d,2 . c̃d,2

C̃

d/2

) /2−1 2 /2 σd ≈ 2e σd . According to [15], c̃d,2 = (2 /2 dσd )−1 and c̃ d,2 = ( d−2 d d,2 Of course, due to the translation invariance of problem, the conditions on moments can be equally imposed on the quantity d

d

d

inf ∫ |x − x0 |γ u0 (x) dx = ∫ |x − x|̄ γ u0 (x) dx,

x0 ∈ℝd

ℝd

ℝd

where x̄ = (∫ℝd xu0 (x) dx)/(∫ℝd u0 (x) dx) is the center of mass of the distribution u0 .

3.1 Solutions blowing up in a finite time

| 75

3.1.1 Proof of blowup involving virial identities Here we present a modification of the classical approach via a moment of a solution, with a power-like weight function, in a more general context of system with two fractional Laplacian operators α

ut + (−Δ) /2 u + ∇ ⋅ (uB(u)) = 0,

(3.1.2)

for (x, t) ∈ ℝd × ℝ+ , where the anomalous diffusion is modeled by a fractional power of the Laplacian, α ∈ (1, 2), and the linear (vector) operator B is defined (formally) as a nonlocal operator β

B(u) = ∇((−Δ)− /2 u).

(3.1.3)

For β ∈ (1, d], and d ≥ 2, one may express the nonlocal nonlinearity in (3.1.2) using convolution operators since B(u)(x) = sd,β ∫ ℝd

x−y u(y) dy, |x − y|d−β+2

(3.1.4)

with some sd,β > 0, and the assumption β > 1 is needed for the convergence of this integral. Of course, the choice α = 2, β = 2 in (3.1.2)–(3.1.4) corresponds to the usual Keller– Segel system. It is well known that if β = 2, the one dimensional system (3.1.2)–(3.1.4) possesses global-in-time solutions not only in the case of classical Brownian diffusion α = 2 but also in the fractional diffusion case 1 < α < 2 and d = 1 as was shown by C. Escudero in [121]. On the other hand, there are many results on the nonexistence of global-in-time solutions with “large” initial data if d ≥ 2 and α = 2, β = 2, see e. g. [148, 15, 18, 163, 69]. Even if d ≥ 2, α = 2 and 1 < β ≤ d, there are results on the blowup of solutions with suitably chosen initial data, see [62] (caution: the notation in [62, Prop. 4.2] differs from that in the present Section). Let us finally recall that, in the limit case α = 2, β = d, mass M = ∫ℝd u0 (x) dx is the critical parameter for the blowup, see [62, Proposition 4.1] and [88]. We note that if the γ-moment ∫ |x|γ u(x) dx is small, the Morrey norms are big. More precisely, we have the following: Lemma 3.1.1 (Comparison of moments and Morrey norms). For each γ > 0, p > 1, there exists a constant C = C(d, p, γ) > 0 such that for every 0 ≤ u ∈ L1loc (ℝd ) |u||M p ≥ C M(

d(1−1/p)/γ

M ) ∫ |x|γ u(x) dx

.

Proof. Evidently, for u ≥ 0 and each R > 0 ∫ |x|γ u(x) dx ≥ Rγ

∫ ℝd \BR (0)

u(x) dx = Rγ (M − ∫ u(x) dx) BR (0)

76 | 3 Blowups holds. To estimate the Morrey M p (ℝd ) norm from below, let us optimize R > 0 in the inequality R /p−d ∫ u(x) dx ≥ R /p−d (M − R−γ ∫ |x|γ u(x) dx) d

d

BR (0)

taking

/γ γ d + γ − d/p ∫ |x| u(x) dx ) , d − d/p M 1

R0 = (

which leads to the inequality in Lemma, see a particular case of γ = 2 in [15, pp. 235– 236]. The usual method of proving the nonexistence of global-in-time nonnegative and nontrivial solutions, used in the above-mentioned papers, consists in the study the evolution of the second moment of a solution w2 (t) = ∫ℝd |x|2 u(x, t) dx, and to show (via suitable differential inequalities) that w2 (t) vanishes for some t > 0 (see also recent paper [100] for moment computations). This is motivated by the kinetic theory origin of the quantities entering into free energy formulation, see Introduction. The second moment of a typical solution to an evolution equation with fractional Laplacian cannot be finite, because of the space asymptotics of solutions to the linear Cauchy problem (22), see Remark 3.1.8 and for example [79]. Hence, our goal in this section is to generalize the classical virial method and to show the blowup of solution system (3.1.2)–(3.1.4) by studying moments of lower-order γ ∈ (1, 2) wγ (t) = ∫ |x|γ u(x, t) dx. ℝd

(3.1.5)

We show the extinction of a moment wγ for some T > 0, which implies limt↗T ‖u(t)‖p = ∞ for every p ∈ (1, ∞], see Remark 3.1.7. Thus, we mean this phenomenon as blowup for (3.1.2). We stress on the fact that it may happen that solution ceases to exist before the extinction of wγ : wγ (T) = 0. In fact, this phenomenon happens for d ≥ 3 since, as a by-product of the result on the initial trace of a nonnegative solution (Remark 5.2.1), d it is known that for each solution its M /2 norm is uniformly bounded, so it does not blow up at the blowup time when ‖u(T)‖∞ = ∞. The authors of [178] showed the blowup of solution to system (3.1.2)–(3.1.4) with fractional diffusion in the particular case d = 2 and β = 2. Our argument is different than that in [178]; shorter, seems to be more direct, and applies in more general situations. Moreover, we are able to formulate a simple condition on the initial data, which leads to the blowup in a finite time of the corresponding solution. We skip the questions of existence of solutions, which are quite standard, referring the readers to the original paper [41]. The crucial role in the approach in this section is played by the following scaling property of system (3.1.2)–(3.1.4):

3.1 Solutions blowing up in a finite time

uλ (x, t) = λα+β−2 u(λx, λα t)

for all

λ > 0,

| 77

(3.1.6)

in the sense that if u is a solution to (3.1.2)–(3.1.4), then uλ is so. In particular, in our construction of solutions to (3.1.2)–(3.1.4), we use the fact that the usual norm of the Lebesgue space Ld/(α+β−2) (ℝd ) is invariant under the transformation u0 (x) 󳨃→ λα+β−2 u0 (λx) for every λ > 0. Theorem 3.1.2. Assume that d ≥ 2, α ∈ (1, 2], and β ∈ (1, d]. Let max{

d 2d , } < p ≤ d. α+β−2 d+β−1

For every u0 ∈ Lp (ℝd ) there exists T = T(‖u0 ‖p ) and the unique local-in-time mild solution u ∈ 𝒞 ([0, T], Lp (ℝd )) of system (3.1.2)–(3.1.4) with u0 as the initial condition. ii) There is ε > 0 such that for every u0 ∈ Ld/(α+β−2) (ℝd ) satisfying i)

‖u0 ‖d/(α+β−2) ≤ ε,

(3.1.7)

there exists a global-in-time mild solution u ∈ 𝒞 ([0, ∞), Lp (ℝd )) of system (3.1.2)– (3.1.4) with u0 as the initial condition. Moreover, if u0 (x) ≥ 0, then the solution u in either i) or ii) above is nonnegative. Finally, if u0 ∈ L1 (ℝd ), then the corresponding solution conserves mass ∫ u(x, t) dx = ∫ u0 (x) dx ≡ M. ℝd

(3.1.8)

ℝd

Recall that condition α > 1 is a usual assumption ([62, Theorem 2.2], [48, 49]), which permits us to control locally the nonlinearity in (3.1.2)–(3.1.4) by the linear term. The results stated above can be easily generalized for equations with general Lévy diffusion operators considered in [48, 49], but we do not pursue this question here. We refer the reader to the above-mentioned papers, as well as [62], for physical motivations to study such equations. On the other hand, motivations stemming from probability theory (propagation of chaos property for interacting particle systems) can be found in e. g. [35]. Remark 3.1.3. For α ∈ (1, 2) and β > 1 satisfying α+β > d+2, it can be shown that localin-time solutions can be continued to the global-in-time ones, see [62, Theorem 3.2]. However, in [62], another approach (via weak solutions) is used to construct solutions of system (3.1.2)–(3.1.4). The proof of Theorem 3.1.2 on local and global solutions to (3.1.2)–(3.1.4) follows a more or less standard reasoning. Our main goal, however, is to prove the finite time blowup of nonnegative solutions to the nonlocal system (3.1.2)–(3.1.4), but a priori even less regular than those in Theorem 3.1.2.

78 | 3 Blowups Theorem 3.1.4. Assume that d ≥ 2. The nonnegative solution of (3.1.2)–(3.1.4) with a nonnegative and nonzero regular initial condition u(x, 0) = u0 (x) cannot exist globally in time in each of the following cases: i) (large mass) for α = 2, β = d, u0 ∈ L1 (ℝd , (1 + |x|2 ) dx), and if M = ∫ u0 (x) dx > ℝd

2d , sd,β

1 so that the threshold with the constant sd,β defined in (3.1.4); in particular, s2,2 = 2π value of M is 8π if d = 2; ii) (high concentration) for α ∈ (1, 2] and β ∈ (1, d] satisfying α + β < d + 2, u0 ∈ L1 (ℝd , (1 + |x|γ ) dx) for some γ ∈ (1, α), and if

∫ℝd |x|γ u0 (x) dx ∫ℝd u0 (x) dx

≤ c( ∫ u0 (x) dx)

γ d+2−α−β

(3.1.9)

ℝd

for certain (sufficiently small) constant c > 0 independent of u0 . The result stated in i) is essentially contained in [88]. The condition for blowup in the form (3.1.9) appeared already in [15] and [194]; of course, for α = 2 only. Note that i) is a limit case of ii). Indeed, (3.1.9) written as (

∫ℝd |x|γ u0 (x) dx ∫ℝd u0 (x) dx

d+2−α−β γ

)

≤ cM,

becomes a condition on (sufficiently large) mass: 1 ≤ cM, when (α + β) ↗ (d + 2). We have to emphasize that Theorem 3.1.4 (ii) contains a result, which is new even for the classical parabolic-elliptic Keller–Segel model (that is, equations (3.1.2)–(3.1.4) with α = β = 2). Indeed, for d ≥ 3, the conditions from part (ii) of Theorem 3.1.4 guarantee the blowup in a finite time if the moment of order γ of the initial condition is finite for some γ ∈ (1, 2). All other known proofs of the blowup required just γ = 2. The following result for d = 2 is also an immediate consequence of Theorem 3.1.4 (ii): Corollary 3.1.5. Assume that α = β = d = 2 in equations (3.1.2)–(3.1.4). Suppose that there exists γ ∈ (1, 2) such that 0 ≤ u0 ∈ L1 (ℝ2 , (1 + |x|γ ) dx). There exists Mγ > 0 such that if ∫ u0 (x) dx > Mγ , ℝ2

then the nonnegative solution of (3.1.2)–(3.1.4) with the initial condition u(x, 0) = u0 (x) ceases to exist in a finite time.

3.1 Solutions blowing up in a finite time

| 79

The assumptions in Corollary 3.1.5 are not optimal: nonnegative solutions to equations (3.1.2)–(3.1.4) with α = β = d = 2 blow up in finite time, under the assumption that their mass is larger than 8π and no moment condition imposed on the initial data is necessary, see Chapter 3, Section 3.4 and [163, Theorem 1.5], [20, Proposition 2.2, Theorem 3.1] in the radially symmetric case for a detailed presentation. Remark 3.1.6. Let us observe that the assumptions (3.1.7) and (3.1.9) for u0 are in a sense complementary due to the following elementary inequality, involving the Lp -norms, mass M = ∫ℝd u(x) dx, and the moment wγ = ∫ℝd |x|γ u(x) dx of a nonnegative function u d

M γ ‖u‖p ≥ CM( ) wγ

(1− p1 )

(3.1.10)

.

Clearly, inequality (3.1.10) is a consequence of Lemma 3.1.1 since the Lp norm is stronger than the M p norm. But a straightforward argument to prove (3.1.10) is that wγ ≥ Rγ ∫ℝd \B (0) u(x) dx, so that R

∫ u(x) dx = M − BR (0)

u(x) dx ≥ M − R−γ wγ .

∫ ℝd \BR (0)

Multiplying both sides of this inequality by Rd(1− /p) , we get with C = ωd1− /p 1

1

1−1/p

1/p

p

C‖u‖p = ( ∫ u (x) dx) ( ∫ dx) ≥R

d( p1 −1)

BR (0)

ℝd d(1/p−1)

R

∫ u(x) dx ≥ Rd( /p−1) M − R−γ wγ . 1

BR (0)

Taking the optimal R, that is, Rγ = Cwγ /M, we get (3.1.10). Now, it is clear that if condition (3.1.9) for blowup is satisfied for some γ ∈ (1, 2] and a suitable constant c, then for p = d/(α + β − 2) appearing in Theorem 3.1.2 (ii), we γ

d

(1− 1 )

have ‖u‖p ≥ CMM d+2−α−β γ p = C, so condition (3.1.7) for global existence is violated for sufficiently small ε > 0. Vice versa, if (3.1.7) is satisfied, (3.1.9) cannot be true with small constants c. −

Remark 3.1.7. We prove Theorem 3.1.4 and Corollary 3.1.5 by showing the extinction in a finite time of the function w(t) = ∫ℝd φγ (x)u(x, t) dx, where φγ (x) is smooth and behaves like |x|γ with some γ ∈ (1, α) for large |x|, see (3.1.11). By (3.1.10) and conservation of mass, this might be interpreted as the blowup of certain norms (in particular, Lp -norms for each p ∈ (1, ∞]) of nonnegative solutions, but we stress on the fact that a nonnegative solution could cease to exist even before the critical time T < ∞, suggesting by (3.1.10) that limt↗T ‖u(t)‖p = ∞.

80 | 3 Blowups Remark 3.1.8. Note that if α < 2, we cannot expect the existence of higher-order moα ments wγ defined in (3.1.5) with γ ≥ α. Indeed, even for the linear equation vt +(−Δ) /2 v = d/α 0, the fundamental solution pα (x, t) behaves like pα (x, t) ≍ (t + |x|d+α /t)−1 , and therefore the moment (3.1.5) with γ ≥ α cannot be finite, see [79] and references given there. Thus, we cannot apply the usual reasoning, which involves an analysis of the evolution of the second moment w2 of the solution, because the integral defining the second moment w2 may diverge. We recall in Proposition 3.6.4 a result showing that the moment (3.1.5) is finite for a large class of initial conditions in the case of γ < α. The main role in our proof of the blowup of solutions to (3.1.2)–(3.1.4) is played by the following smooth nonnegative weight function on ℝd : γ/2

φ(x) = φγ (x) ≡ (1 + |x|2 ) − 1

(3.1.11)

with γ ∈ (1, 2]. Since (1 + |x|2 )γ ≤ (1 + |x|γ )2 , we have for each ε > 0, suitably chosen C(ε) > 0, and for every x ∈ ℝd φ(x) ≤ |x|γ ≤ ε + C(ε)φ(x).

(3.1.12)

Next, let us state two auxiliary results concerning the weight function φ, which will be used in the proof of Theorem 3.1.4. Here, for a given φ ∈ 𝒞 2 (ℝd ), we denote by D2 φ its Hessian matrix. Moreover, the scalar product of vectors x, y ∈ ℝd is denoted by x ⋅ y. If A is either a vector or a matrix, the expression |A| means its Euclidean norm. Lemma 3.1.9. Let α ∈ (1, 2), γ ∈ (1, α), and φ be defined by (3.1.11). Then (−Δ) /2 φ ∈ L∞ (ℝd ). α

(3.1.13)

Proof. First, note that by a direct computation, we have γ/2−1

∇φ(x) = γ(1 + |x|2 )

x,

(3.1.14)

and γ/2−2

𝜕xj 𝜕xi φ(x) = (γ(1 + |x|2 )δij − γ(2 − γ)xi xj )(1 + |x|2 )

.

(3.1.15)

In particular, for every R > 0, there exists C(R, γ) > 0 such that for all |x| ≥ R we have 󵄨 󵄨󵄨 γ−1 󵄨󵄨∇φ(x)󵄨󵄨󵄨 ≤ C(R, γ)|x|

󵄨 󵄨 and 󵄨󵄨󵄨D2 φ(x)󵄨󵄨󵄨 ≤ C(R, γ)|x|γ−2 .

(3.1.16)

Now, we apply the following Lévy–Khintchine integral representation of the fractional Laplacian (23): α

(−Δ) /2 φ(x) = C(d, α) ∫ ℝd

φ(x + y) − φ(x) − ∇φ(x) ⋅ y dy |y|d+α

(3.1.17)

3.1 Solutions blowing up in a finite time

| 81

(with a suitable constant C(d, α)), which is valid for every α ∈ (1, 2), see [115, Theorem 1] for a detailed proof of that version of Lévy–Khintchine formula. Using the Tayα lor expansion and estimates (3.1.16), one can immediately show that (−Δ) /2 φ(x) is well α defined for every x ∈ ℝd and, moreover, sup|x|≤R |(−Δ) /2 φ(x)| < ∞ for each R > 0. To obtain an estimate uniform in x ∈ ℝd , we assume that |x| ≥ 1, and we shall estimate the integral on the right-hand side of (3.1.17) for |y| ≤ |x|/2 and |y| > |x|/2, separately. If |y| ≤ |x|/2, by the Taylor formula and the second inequality in (3.1.16), we obtain 1

󵄨 1 2 󵄨 2 󵄨󵄨 󵄨 󵄨󵄨φ(x + y) − φ(x) − ∇φ(x) ⋅ y󵄨󵄨󵄨 ≤ |y| ∫󵄨󵄨󵄨D φ(x + sy)󵄨󵄨󵄨 ds 2 0

2

1

≤ C|y| ∫ |x + sy|γ−2 ds. 0

Since |y| ≤ |x|/2 and s ∈ [0, 1], we can estimate 󵄨 1 󵄨 󵄨 󵄨 |x + sy| ≥ 󵄨󵄨󵄨|x| − s|y|󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨|x| − |y|󵄨󵄨󵄨 ≥ |x|. 2 Consequently, for γ − 2 < 0, we obtain 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

∫

{|y|≤|x|/2}

φ(x + y) − φ(x) − ∇φ(x) ⋅ y 󵄨󵄨󵄨󵄨 dy󵄨󵄨 󵄨󵄨 |y|d+α ≤ C|x|γ−2

dy

∫ {|y|≤|x|/2}

|y|d+α−2

= C|x|γ−α

(3.1.18)

for all |x| ≥ 1 and a constant C > 0 independent of x. If |y| ≥ |x|/2 and |x| ≥ 1, we combine the first inequality from (3.1.16) (remember that γ − 1 > 0) with the Taylor expansion to show 1

󵄨 󵄨󵄨 󵄨 󵄨 γ−1 γ−1 󵄨󵄨φ(x + y) − φ(x)󵄨󵄨󵄨 ≤ |y| ∫󵄨󵄨󵄨∇φ(x + sy)󵄨󵄨󵄨 ds ≤ C|y|(|x| + |y| ). 0

Hence, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

∫ {|y|>|x|/2}

φ(x + y) − φ(x) − ∇φ(x) ⋅ y 󵄨󵄨󵄨󵄨 dy󵄨󵄨 󵄨󵄨 |y|d+α

≤ C(|x|γ−1

∫ {|y|>|x|/2}

dy + |y|d+α−1

∫ {|y|>|x|/2}

dy ) = C|x|γ−α |y|d+α−γ

(3.1.19)

for all |x| ≥ 1 and a constant C > 0 independent of x. Finally, application of inequalities (3.1.18) and (3.1.19) completes the proof, because γ < α.

82 | 3 Blowups Remark 3.1.10. Note that above, we have, in fact, proved that α 󵄨 󵄨 sup (1 + |x|α−γ )󵄨󵄨󵄨(−Δ) /2 φ(x)󵄨󵄨󵄨 < ∞

for every γ ∈ (1, α).

x∈ℝd

Lemma 3.1.11. For every γ ∈ (1, 2], the function φ—defined in (3.1.11)—is uniformly convex on compact subsets of ℝd . Moreover, there exists K = K(γ) such that the inequality (∇φ(x) − ∇φ(y)) ⋅ (x − y) ≥

K|x − y|2 1 + |x|2−γ + |y|2−γ

(3.1.20)

holds true for all x, y ∈ ℝd . Proof. Using the explicit expression for the Hessian matrix of φ in (3.1.15), we obtain 2

D φ(x)y ⋅ y =

γ(1 + |x|2 )|y|2 − γ(2 − γ)(∑i xi2 yi2 + ∑i=j̸ xi xj yi yj )

(3.1.21)

(1 + |x|2 )2−γ/2

for every x, y ∈ ℝd . Now, by the elementary inequality xi xj yi yj ≤ immediately obtain

1 2 2 (x y 2 i j

+ xi2 yj2 ), we

∑ xi xj yi yj ≤ ∑ xi2 yj2 . i=j̸

i=j̸

Consequently, ∑ xi2 yi2 + ∑ xi xj yi yj ≤ ∑ xi2 yj2 = |x|2 |y|2 . i

i,j

i=j̸

(3.1.22)

Since γ ∈ (1, 2], applying estimate (3.1.22) to (3.1.21), we get the inequality D2 φ(x)y ⋅ y ≥

γ(1 + (γ − 1)|x|2 )|y|2 , (1 + |x|2 )2−γ/2

which leads directly to the estimate from below D2 φ(x)y ⋅ y ≥

(γ − 1)|y|2 . (1 + |x|2 )1−γ/2

(3.1.23)

Finally, it follows from the integration of the second derivative of φ that 1

(∇φ(x) − ∇φ(y)) ⋅ (x − y) = ∫ D2 φ(x + s(y − x))(x − y) ⋅ (x − y) ds. 0

Hence, using inequality (3.1.23) and the estimate 󵄨2 1−γ/2 󵄨 ≤ C(1 + |x|2−γ + |y|2−γ ), (1 + 󵄨󵄨󵄨x + s(y − x)󵄨󵄨󵄨 ) valid for all x, y ∈ ℝd , s ∈ [0, 1], and a constant C > 0 independent of x, y, s, one can easily complete the proof of Lemma 3.1.11.

3.1 Solutions blowing up in a finite time

| 83

Proof of Theorem 3.1.4. We consider the moment w = w(t) ≡ ∫ φ(x) u(x, t) dx, ℝd

where the function φ is defined in (3.1.11) and 1 < γ < α. Note that, in view of inequalities (3.1.12), the quantity w is essentially equivalent to the moment wγ of order γ of the solution u. Moreover, it satisfies the relation d α w = − ∫ (−Δ) /2 u(x, t) φ(x) dx − ∫ u(x, t)Bu(x, t) ⋅ ∇φ(x) dx dt ℝd

ℝd

α/2

= − ∫ (−Δ) φ(x) u(x, t) dx ℝd

−

sd,β 2

∫ ∫ (∇φ(x) − ∇φ(y)) ⋅ (x − y) ℝd

ℝd

u(x, t)u(y, t) dx dy |x − y|d−β+2

(3.1.24)

after using the definition of the form B in (3.1.4) and the symmetrization of the double integral. This computation resembles the usual proof of blowup, involving the second moments, see [15, 62, 108, 88]. i) For α = 2 = γ (hence, for φ(x) = |x|2 ) and for β = d, the equality (3.1.24) can be rewritten as follows: d w(t) = 2dM − sd,β M 2 . dt Evidently, for M > 2d/sd,β , this implies the inequality w(T) < 0 for some 0 < T < ∞, a contradiction with the global existence of nonnegative solutions. Thus, we recover the result in [62, Proposition 4.1] refined in [108, 88]. ii) For 1 < β ≤ d and fixed M > 0, we are going to use the following simple identity: M 2 = ∫ ∫ u(x, t)u(y, t) dx dy ℝd ℝd

= ∫ ∫ u(x, t)u(y, t) ℝd ℝd

|x − y|ν (1 + |x|2−γ + |y|2−γ )δ dx dy 2−γ 2−γ δ |x − y|ν (1 + |x| + |y| )

with some ν > 0 and δ > 0. Now, we apply the Hölder inequality with the exponents p chosen so that p > 1 and p󸀠 = p−1 νp = d − β,

δp = 1,

νp󸀠 + (2 − γ)δp󸀠 = γ.

(3.1.25)

Of course, such a choice of ν, δ, p is possible whenever β < d and γ < 2, because we only need d − β + 2 − γ = γ(p − 1). If β = d, it suffices to take ν = 0 and p = 2/γ > 1. As a

84 | 3 Blowups consequence, we get M 2 ≤ J(t) /p 1

νp󸀠

× ( ∫ ∫ u(x, t)u(y, t)|x − y|

2−γ

(1 + |x|

+ |y|

󸀠 2−γ δp

)

1/p󸀠

dx dy) ,

(3.1.26)

ℝd ℝd

where the integral J(t) satisfies J(t) = ∫ ∫ ℝd ℝd

≤

u(x, t)u(y, t) dx dy |x − y|d−β 1 + |x|2−γ + |y|2−γ

u(x, t)u(y, t) 1 dx dy ∫ ∫ (∇φ(x) − ∇φ(y)) ⋅ (x − y) K |x − y|d−β+2

(3.1.27)

ℝd ℝd

by Lemma 3.1.11. It follows from relations (3.1.25) and inequalities (3.1.12) that there exists a constant C1 > 0 such that |x − y|νp (1 + |x|2−γ + |y|2−γ ) 󸀠

δp󸀠

≤ C1 (1 + φ(x) + φ(y)).

(3.1.28)

Hence, (3.1.26) implies 1/p󸀠

M 2 ≤ C1/p J(t) /p (M 2 + 2Mw(t)) . 1 󸀠

1

(3.1.29)

Going back to identity (3.1.24), we obtain from Lemma 3.1.9 and from inequalities (3.1.27)–(3.1.29) that M 2p d w(t) ≤ C2 M − C3 2 󸀠 dt (M + 2Mw(t))p/p

(3.1.30)

α

with C2 = ‖(−Δ) /2 φ‖∞ and a suitable constant C3 > 0. Now, we fix for a while M = M0 in (3.1.30) so large in order to have C2 M0 − C3

M02p

(M02 )p/p

󸀠

< 0.

(3.1.31)

Hence, there exists C4 = C4 (M0 ) > 0 such that for 0 < w(0) ≤ C4 we still have C2 M 0 − C3

M02p

(M02 + 2M0 w(0))p/p

󸀠

< 0.

It is clear that if initially 0 < w(0) ≤ C4 , then, by inequality (3.1.30) with M = M0 , the function w(t) is decreasing in time. Moreover, M02p d w(t) ≤ C2 M0 − C3 2 󸀠 < 0 dt (M0 + 2M0 w(0))p/p

3.2 A novel approach to blowup and concentration of mass in two dimensions | 85

and, consequently, w(T) < 0 for some 0 < T < ∞. This contradicts the global-in-time existence of regular nonnegative solutions of (3.1.2)–(3.1.4). Finally, note that due to the first inequality in (3.1.12), it suffices to assume w(0) ≤ ∫ |x|γ u0 (x) dx ≤ C4

and

∫ u0 (x) dx = M0 ,

(3.1.32)

ℝd

ℝd

to obtain the blowup in a finite time of the corresponding solution. Now, assume that ∫ℝd u(x, t) dx = ∫ℝd u0 (x) dx = M ≠ M0 . Recall that system

(3.1.2)–(3.1.4) is invariant under the scaling (3.1.6). Choosing λα+β−2−d = M0 /M, we obtain ∫ℝd uλ (x, t) dx = ∫ℝd uλ0 (x) dx = M0 and, by (3.1.32), the blowup of the solution takes place under the assumption ∫ |x|γ uλ0 (x) dx ≤ C4 . ℝd

Changing the variables and using the explicit form of λ, we obtain the blowup of solutions to (3.1.2)–(3.1.4) under the following assumption on the initial condition: γ

∫ |x|γ u0 (x) dx ≤ C4 M0

−1+ α+β−2−d

ℝd

γ

1+ d+2−α−β

( ∫ u0 (x) dx)

.

ℝd

Proof of Corollary 3.1.5. We follow the proof of Theorem 3.1.4. In particular, we choose Mγ so large that the inequality (3.1.31) holds true for all M0 > Mγ . This leads to the blowup of the corresponding solution under the assumption (3.1.32) imposed on the initial data. To complete the proof, we use the scaling argument again. By (3.1.6), uλ (x, t) = 2 λ u(λx, λ2 t) is a solution for every λ > 0. Note now that ∫ uλ0 (x) dx = ∫ u0 (x) dx and ∫ |x|γ uλ0 (x) dx = λ−γ ∫ |x|γ u0 (x) dx. ℝ2

ℝ2

ℝ2

ℝ2

Hence, each initial data u0 ∈ L1 (ℝ2 , (1 + |x|γ ) dx) satisfying ∫ℝ2 u0 (x) dx = M0 > M γ leads to the blowup in a finite time of the corresponding solution because the moment condition in (3.1.32) can be satisfied replacing u by uλ and choosing λ large enough.

3.2 A novel approach to blowup and concentration of mass in two dimensions A simple proof of concentration of mass equal to 8π for blowing up N-symmetric solutions of the Keller–Segel model of chemotaxis in two dimensions with large N is given.

86 | 3 Blowups In this situation, quantization of mass at blowup time occurs with 8kπ, k = 1, only. The presentation here is adopted from [50]. A criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is derived for radial solutions of chemotaxis in several dimensions. This condition is, in a sense, complementary to the one guaranteeing the global-in-time existence of solutions. We once more consider in this section the classical parabolic-elliptic Keller–Segel model of chemotaxis in d ≥ 2 space dimensions ut − Δu + ∇ ⋅ (u∇v) = 0,

Δv + u = 0,

(3.2.1) (3.2.2)

supplemented with a nonnegative initial condition u(x, 0) = u0 (x) ≥ 0.

(3.2.3)

Here, for (x, t) ∈ ℝd × [0, T), the function u = u(x, t) ≥ 0 denotes the density of the population of microorganisms; v = v(x, t), the density of the chemical secreted by themselves that attracts them and makes them to aggregate. The system (3.2.1)–(3.2.2) also models the gravitational attraction of particles in astrophysical models, see [16, 15]. As it is well known, see [69, 7], the total mass of the initial condition M = ∫ u0 (x) dx,

(3.2.4)

ℝd

conserved in time, is the critical quantity for the global-in-time existence of nonnegative solutions in the two dimensional case. Namely, if M ≤ 8π, then solutions of (3.2.1)–(3.2.3) (with u0 —a finite nonnegative measure) exist for all t ≥ 0. For the localin-time existence, it should be assumed that all the atoms of the finite measure u0 are of mass less than 8π, see [69, 7, 64]. When M > 8π, nonnegative solutions blow up in a finite time, and for radially symmetric solutions mass equal to 8π concentrates at the origin at the blowup time, see [226, Chapter 11], [20], respectively. After the blowup, the continuation of solutions as measured valued functions of time is, in general, not unique (see [113, 182]). In the radially symmetric case, however, such a continuation leads to the concentration of the whole mass M at the origin in the infinite time, see [20]. The multidimensional case is different: there are solutions of the chemotaxis system with arbitrarily small M > 0 that cease to exist after a finite time elapsed, see for instance [15, 139, 41]. These issues will be also discussed in Chapter 5, Section 5.6. First, we show in the present work that the radial concentration of data is the critical quantity for the finite time blowup of nonnegative radial solutions of (3.2.1)– (3.2.3). In this section, we define the ψ-radial concentration by the formula x ||u0 ||ψ ≡ sup R2−d ∫ ψ( )u0 (x) dx R R>0 {|x|≤R}

(3.2.5)

3.2 A novel approach to blowup and concentration of mass in two dimensions | 87

with a fixed radial nonnegative, piecewise 𝒞 2 function ψ supported on the unit ball, such that ψ(0) = 1. Clearly, those quantities for such weight functions ψ are comparable, so we fix in ψ(x) = (1 − |x|2 )2+ , see (3.3.2) below. d Of course, for d ≥ 3 and p = d/2, the norm in M /2 (ℝd ) (relevant to the theory of existence of local-in-time solutions) dominates the radial concentration (3.2.5): |u0|M d/2 ≥

R2−d ∫{|x|≤R} u0 (x) dx for each R > 0, but, in fact, for radially symmetric u0 both quantities |u0|M d/2 and ||u0 ||ψ are equivalent, see also Proposition 2.7.1. The criticality of the radial concentration (3.2.5) means that for initial data with small ||u0 ||ψ solutions exist indefinitely in time (see [16, 172]), whereas our result stated in Theorem 3.2.1 shows that for sufficiently big ||u0 ||ψ regular solutions cease to exist in a finite time. We recall that other blowup criteria appeared in [15] and [41]. They have been, however, formulated in terms of “global quantities” like the second moment ∫ |x|2 u0 (x) dx, whereas (3.2.5) is a local quantity, and its definition does not require supplementary properties of u0 , such as ∫ |x|2 u0 (x) dx < ∞. The proof of our first result (contained in the following theorem) on the occurrence of radially symmetric blowup for d ≥ 2 does not involve global quantities, and its idea is astonishingly simple. Theorem 3.2.1 (Blowup of radial solutions with large concentration). For each d ≥ 2, there exists a constant Cd > 0 such that if u0 ∈ L1 (ℝd ) is a radially symmetric function and R2−d ∫{|x|≤R} ψ( Rx )u0 (x) dx > Cd for some R > 0, then the solution u of problem (3.2.1)– (3.2.3) blows up in a finite time. Remark 3.2.2. Note that for d = 2, we recover the well known result: if M > C2 = 8π, then the solution of (3.2.1)–(3.2.3) blows up in a finite time, see the end of the proof of Theorem 3.2.1. In fact, this proof (involving a local moment of the solution) extends to the two dimensional case x ∈ ℝ2 of arbitrary (not necessarily radially symmetric) nonnegative solutions, see [194, 163, 28] for similar arguments. Some further improvements of the results in Theorem 3.2.1 (with direct relations to the critical values of Morrey norms and with quite different proofs) are in Sections 3.2 and 3.5 (see [50, 28, 51]). In our second result, we limit ourselves to the Cauchy problem (3.2.1)–(3.2.3) on the plane, and we show a concentration at the origin of mass equal exactly to 8π for some solutions. This result is known in the radially symmetric case; our proof allows us to deal with a larger class of solutions, and is conceptually much simpler than existing ones. An analysis of the chemotaxis system in bounded planar domains leads to blowups at interior and boundary points, see [226, Theorem 1.1], and to the quantization of mass 8kπ, k ∈ ℕ, at the interior blowup points for solutions with finite free energy, see [226, Theorem 1.2, Theorem 15. 1]. The proofs of those results in [226, Chapters 11–15] rely on subtle estimates of the free energy for (3.2.1)–(3.2.2) and various functional inequalities of Gagliardo–Nirenberg–Sobolev type. For an earlier approach to different versions of conjectures on concentration, we refer the reader to [222, p. 23–24]. Compare also [20] for the radially symmetric case.

88 | 3 Blowups Our proof for radially symmetric solutions with the initial condition satisfying u0 (zeiϑ ) = u0 (z),

z ∈ ℂ,

ϑ ∈ ℝ,

(3.2.6)

extends to the case of N-symmetric solutions with sufficiently large N, that is, those with the initial data satisfying u0 (zeik

2π/N

) = u0 (z),

z ∈ ℂ,

k ∈ ℕ,

(3.2.7)

with the natural identification ℝ2 ∋ x ↔ z ∈ ℂ. Note that by the uniqueness of nonnegative solutions, the solution u(x, t) with u0 satisfying (3.2.6) is radial, and for that 2π satisfying (3.2.7) u is N-symmetric for each admissible t: u(zeik /N , t) = u(z, t). The interest in such solutions is related to the problem of studying certain bilinear integrals, involving derivatives of the fundamental solution of Laplacian. An extension of these results for initial distributions close to N-symmetric (in a sense of, say, small variation of the difference of a given distribution and an N-symmetric distribution) is also possible. Moreover, we are motivated by results in [219], where 2-symmetric distributions have been considered, see Remark 3.2.5 for more information. Theorem 3.2.3 (Blowup with concentration of mass equal to 8π). Assume that the initial condition 0 ≤ u0 ∈ L1 (ℝ2 ) is N-symmetric in the sense of equation (3.2.7), and such that ∫ u0 (x) dx = M > 8π. Let u(x, t), for x ∈ ℝ2 and t < Tb , be the corresponding classical solution of problem (3.2.1)–(3.2.3), which cannot be continued past the blowup time t = Tb . If N is sufficiently large so that M/N is small enough, then lim sup

t→Tb , R→0

∫ u(x, t) dx ≤ 8π. {|x|≤R}

In fact, N-symmetric solutions blow up with the concentration of mass equal to 8π. Corollary 3.2.4. Under the assumptions of Theorem 3.2.3, if moreover, the moment ∫ |x|2 u0 (x) dx is finite, then lim lim

R→0 t→Tb

∫ u(x, t) dx = 8π. {|x|≤R}

In the proof of Theorem 3.2.3, we use simple (but rather subtle) techniques of weight functions and scalings. The core of our analysis consists in uniform (with respect to initial data) estimates on a blowup time (see Proposition 3.4.1) and on the uniform spread (or decay) of mass for symmetric initial conditions (see Proposition 3.4.8). The proofs of these two propositions are much shorter in the radially symmetric case,

3.3 Proof of blowup of radial solutions | 89

which we emphasize below. Moreover, we use systematically the well known rescaling of the system: for each λ > 0 and each solution u of (3.2.1)–(3.2.2) of mass M the function uλ (x, t) = λ2 u(λx, λ2 t)

(3.2.8)

is also a solution, with its mass equal to M, as well. Corollary 3.2.4 is a direct consequence of Theorem 3.2.3 combined with results proved in [64], see the end of Section 3.4. Remark 3.2.5. The authors of [219] suggested how to construct solutions of the Keller– Segel system that blow up with the quantized concentration of mass M = 16π. In view of Theorem 3.2.3, their data cannot be N-symmetric with large N ≫ 2.

3.3 Proof of blowup of radial solutions We begin with two elementary observations, which will be used in the proof of Theorem 3.2.1. The first is again Lemma 2.1.8, which permits us to estimate the nonlinear term; the second is Lemma 3.3.1. If ω ∈ L1loc (ℝd ) is a radially symmetric function and M(R) = ∫{|x|≤R} ω(x) dx its radial distribution function, then 1 ∫ ω(x)M(|x|) dx = M(R)2 . 2

{|x|≤R}

Proof. Since ω is radial, it satisfies for |x| = R the equality ω(x) = using the polar coordinates, we obtain R

∫ ω(x)M(|x|) dx = σd ∫ 0

{|x|≤R}

1 1−d 󸀠 R M (R). σd

Thus,

1 1−d 󸀠 r M (r)M(r)r d−1 dr σd

R

1 = ∫ M 󸀠 (r)M(r) dr = M(R)2 . 2 0

Proof of Theorem 3.2.1. We will derive a differential inequality for the quantity wR (t) = ∫ ψR (x)u(x, t) dx

(3.3.1)

with the scaled weight function ψR supported on the ball {|x| ≤ R} 2

ψ(x) = (1 − |x|2 ) 1{|x|≤1}

x and ψR (x) = ψ( ) R

with R > 0.

(3.3.2)

90 | 3 Blowups The function ψ ∈ 𝒞 1 (ℝd ) has piecewise continuous and bounded second derivatives ∇ψ(x) = −4x(1 − |x|2 )1{|x|≤1} = −4x ψ(x) /2 , 1

Δψ(x) = (−4d + 4(d + 2)|x|2 )1{|x|≤1} .

(3.3.3)

Observe that ψ satisfies the relation Δψ(x) ≥ −

(d + 2)2 ψ(x), 2

(3.3.4)

which is seen from the elementary inequality for the quadratic polynomial −4d + 4(d + 2)s ≥ −

(d + 2)2 (1 − s)2 , 2

)2 ≥ 0, applied to 0 ≤ s ≤ 1. equivalent to (s − d−2 d+2 Now, using equation (3.2.1), integrations by parts and applying relations (3.3.2)– (3.3.4), we obtain d w (t) = ∫ ΔψR (x)u(x, t) dx + ∫ u(x, t)∇v(x, t) ⋅ ∇ψR (x) dx dt R (d + 2)2 ≥ R−2 (− ∫ ψR (x)u(x, t) dx 2 1/2

− 4 ∫ u(x, t)(∇v(x, t) ⋅ x)(ψR (x)) dx) .

(3.3.5)

Thus, by Lemma 2.1.8, we get R2

(d + 2)2 d wR (t) ≥ − wR (t) dt 2 4 1 + ∫ u(x, t)M(|x|, t)|x|2−d ψR (x) /2 dx σd ≥−

(3.3.6)

(d + 2)2 4 wR (t) + R2−d ∫ ψR (x)u(x, t)M(|x|, t) dx, 2 σd 1

because ψR (x) = 0 for |x| ≥ R, and ψR (x) ≤ ψR (x) /2 . Now, note that, obviously, M(R, t) =

∫ u(y, t) dy ≥ {|y|≤R}

∫ ψR (y)u(y, t) dy. {|y|≤R}

Hence, applying Lemma 3.3.1 to the radial function ω(x) = ψR (x)u(x, t), we obtain 2

1 ∫ ψR (x)u(x, t)M(|x|, t) dx ≥ (∫ ψR (x)u(x, t) dx) . 2

3.4 Blowup in the plane ℝ2 with 8π concentration of mass |

91

Thus, as a consequence of inequality (3.3.6), we arrive at R2

d (d + 2)2 2 wR (t) ≥ − wR (t) + R2−d wR (t)2 . dt 2 σd

(3.3.7)

Now, it is clear from (3.3.7) that if R2−d wR (0) > 2

wR (0) + then − (d+2) 2

σ (d + 2)2 2 / ( ) = (d + 2)2 d ≡ Cd , 2 σd 4

2 2−d R wR (0)2 σd

≡ δ > 0. Since the right-hand side of (3.3.7) is an

d increasing function of wR , we have dt wR (t) ≥ δ > 0. As a consequence, the function wR (t) becomes greater than M = ∫ u(x, t) dx in a finite time, which is a contradiction with the existence of nonnegative, mass conserving solutions. Finally, observe that if d = 2, the conditions M > C2 ≡ 8π, ||u0 ||ψ > 8π and wR (0) > 8π for R > 0 sufficiently large are equivalent. Similarly, if d ≥ 3, the conditions ||u0 ||ψ > Cd and R2−d wR (0) > Cd for some R > 0 are equivalent.

3.4 Blowup in the plane ℝ2 with 8π concentration of mass The proof of Theorem 3.2.3, saying that a solution to problem (3.2.1)–(3.2.3) on the whole plane ℝ2 with M > 8π concentrates at the origin with mass not exceeding 8π at the blowup time, is based on two auxiliary results: on a uniform estimate of the blowup time in Proposition 3.4.1, and on a uniformly slow spread of mass over annuli in ℝ2 in Proposition 3.4.8. 3.4.1 Uniform blowup time In the following proposition, we show that the blowup time of a solution to problem (3.2.1)–(3.2.3) can be estimated from above by a number, which depends only on an amount of u0 concentrated in the unit ball. Proposition 3.4.1. Let ε > 0 and γ > 0 be arbitrary and fixed. Suppose that u = u(x, t) is a solution of problem (3.2.1)–(3.2.3) with an initial datum satisfying 0 ≤ u0 ∈ L1 (ℝ2 ) and

∫ u0 (x) dx ≥ 8π + ε. {|x|≤γ}

Then u(x, t) blows up in a finite time t = Tb ≤ γ 2 T(M, ε), where T(M, ε) > 0 depends on M = ∫ u0 (x) dx > 8π and ε, only.

92 | 3 Blowups Remark 3.4.2. We introduce the parameter γ > 0 in Proposition 3.4.1 to simplify the notation in the proof. In fact, assuming the property stated in the proposition for γ = 1, we obtain immediately this property for each other γ > 0 by the rescaling u 󳨃→ uγ−1 = γ −2 u(γ −1 x, γ −2 t). Remark 3.4.3. Observe that for an initial condition u0 with its support in the unit ball, this proposition holds true by the standard second-moment argument (see [15, 69]) based on the identity d 1 (∫ |x|2 u(x, t) dx) = M(8π − M) < 0, dt 2π which implies that a nonnegative solution u(x, t) ceases to exist at a moment of time estimated from above by the number 2π(M(M − 8π))−1 ∫ |x|2 u0 (x) dx. Now, it suffices to notice that ∫ |x|2 u0 (x) dx ≤ M for supp u0 ⊂ {|x| ≤ 1}, and choose ε = M − 8π. Remark 3.4.4. Proposition 3.4.1 has been already proved in this chapter in the radially symmetric case. Indeed, it is sufficient to apply inequality (3.3.7) with d = 2 and a suitable R > 0: d 1 w (t) ≥ −8wR (t) + wR (t)2 dt R π

R2

(3.4.1)

to the function wR defined by relations (3.3.1)–(3.3.2). We use this inequality with R = 1 1 2 /2 (2 + 16π/ε) /2 . By a direct calculation using the assumption on u0 , we obtain wR (0) ≥ (1 −

2

1 ) R2

∫ u0 (x) dx ≥ (1 − {|x|≤1}

2

ε 1 ) (8π + ε) ≥ 8π + . 2 R2

Thus, analogously as at the end of the proof of Theorem 3.2.1, the function wR (t) becomes greater than M in a finite time, which can be estimated from above by a quantity depending on M and ε, only. Remark 3.4.5. We cannot directly apply the local moment method developed in [28] to show Proposition 3.4.1 for general initial conditions, analogously as in the radial case discussed in Remark 3.4.4. This is due to the fact that blowup results in [28] are proved for each initial datum u0 such that M = ∫ u0 (x) dx > 8π, which, moreover, has a small mass outside a ball. In fact, by methods of [64], we can remove that extra assumption from results proved in [28]. One may summarize Remarks 3.4.3–3.4.5 by saying that the main problem in proving Proposition 3.4.1 consists in controlling a large mass of a solution, which is outside of the unit ball. To show this proposition, we study (as in the previous section) the time evolution of the moment with the weight function ψ w(t) = w1 (t) = ∫ ψ(x)u(x, t) dx,

3.4 Blowup in the plane ℝ2 with 8π concentration of mass |

93

and a solution u(x, t) blows up at certain Tb if there exists T ≥ Tb such that w(T) = M. Here, besides inequalities (3.3.3)–(3.3.4), we will use the following elementary estimates for the weight function (3.3.2): 󵄨 󵄨󵄨 2 󵄨󵄨ψ(x) − 1󵄨󵄨󵄨 ≤ B|x| , 󵄨 󵄨󵄨 󵄨󵄨∇ψ(x) − ∇ψ(y) + 4(x − y)󵄨󵄨󵄨 ≤ Bδ|x − y| for all |x|, |y| ≤ δ, 󵄨 󵄨󵄨 2 󵄨󵄨(x − y) ⋅ (∇ψ(x) − ∇ψ(y))󵄨󵄨󵄨 ≤ B min{|x − y| , |x − y|},

(3.4.2) (3.4.3) (3.4.4)

valid for each fixed constant 0 < δ < 1, some constant B ≥ 1 independent of δ, and all x, y ∈ ℝ2 . First, let us prove an auxiliary result concerning the function w(t). Lemma 3.4.6. Given ε ∈ (0, M − 8π], define the parameters η = η(ε) =

ε , 100M 2 B

α=

1 , 100MB

λ = λ(ε) =

100M 2 B + 1, ε

(3.4.5)

where B ≥ 1 is a constant satisfying (3.4.2)–(3.4.4). Assume that ε w(0) ≥ 8π + . 2

(3.4.6)

Suppose that there exists T ∈ (0, Tb ] such that for all t ∈ [0, T) we have the estimate ∫

u(x, t) dx < εα.

(3.4.7)

{η≤|x|≤λ}

Then, for all t ∈ [0, T], the inequality T ≤ 100M/ε follows.

d w(t) dt

≥

1 ε holds true. In particular, the estimate 100

Proof. Applying inequality (3.4.4) and then assumption (3.4.7), we have for our choice of η, α, λ ∫

∫

u(x, t)u(y, t)

{|y|≤η} {η≤|x|≤λ}

|(x − y) ⋅ (∇ψ(x) − ∇ψ(y))| ε , dx dy ≤ BMεα = 100 |x − y|2

(3.4.8)

and again by (3.4.4) since η < 1 ∫

∫ u(x, t)u(y, t)

{|y|≤η} {λ≤|x|}

|(x − y) ⋅ (∇ψ(x) − ∇ψ(y))| M2B ε dx dy ≤ = . λ − 1 100 |x − y|2

Moreover, by (3.4.3) and elementary calculations ∫

∫ u(x, t)u(y, t)

{|y|≤η} {|x|≤η}

(x − y) ⋅ (∇ψ(x) − ∇ψ(y)) dx dy |x − y|2

(3.4.9)

94 | 3 Blowups ≥ w(t)2 − 2w(t)

∫

ψ(x)u(x, t) dx −

{η≤|x|≤1}

1 ε 25

1 ε 25 1 1 1 ε − ε ≥ w(t)2 − ε. = w(t)2 − 50 25 10 ≥ w(t)2 − 2Mεα −

Hence, repeating the above estimate with x replaced by y, and using bound (3.3.4) for Laplacian of weight function with d = 2, we obtain d 1 x−y ⋅ ∇ψ(x) dx dy w(t) = ∫ ψ(x)Δu(x, t) dx − ∬ u(x, t)u(y, t) dt 2π |x − y|2 = ∫ Δψ(x)u(x, t) dx −

1 x−y dx dy ⋅ (∇ψ(x) − ∇ψ(y)) dx dy ∬ u(x, t)u(y, t) 4π |x − y|2

1 ε (w(t)2 − ) π 10 1 1 1 1 1 = w(t)(w(t) − 8π) − ε≥ ε− ε≥ ε, π 10π 2π 10π 4π ≥ −8w(t) +

as long as w(t) is increasing. Now, Lemma 3.4.6 follows by assumption (3.4.6). Since we cannot have the estimate w(T) > M, integrating the differential inequal1 d w(t) ≥ 100 ε, we obtain the upper bound T ≤ 100M/ε. ity dt Now, we define a certain property of problem (3.2.1)–(3.2.3), which will be called the property ℐε . Fix ε > 0 and γ > 0. Problem (3.2.1)–(3.2.3) is said to have the property ℐε if each of its solutions corresponding to an initial datum satisfying 0 ≤ u0 ∈ L1 (ℝ2 ) and

∫ u0 (x) dx ≥ 8π + ε

(3.4.10)

{|x|≤γ}

blows up not later than at time γ 2 T(M, ε), with the parameter T(M, ε) > 0 depending on M = ∫ u0 (x) dx > 8π and ε, only. Let us first notice elementary facts concerning the property ℐε . Remark 3.4.7. The parameter γ > 0 can be easily removed from this definition by the usual rescaling, see Remark 3.4.2. Problem (3.2.1)–(3.2.3) has the property ℐM−8π , because then assumptions (3.4.10) with ε = M − 8π mean that u0 is supported on the ball of radius γ. Hence, it suffices to apply Remark 3.4.3. Obviously, there is no solution satisfying conditions (3.4.10) for ε > M − 8π. It is also easy to show that if problem (3.2.1)–(3.2.3) has the property ℐε , then it has the property ℐε̃ for each ε̃ > ε. Proof of Proposition 3.4.1. By definition of the property ℐε , it suffices to show that problem (3.2.1)–(3.2.3) has the property ℐε for all ε > 0, and with a suitably chosen γ > 0. To

3.4 Blowup in the plane ℝ2 with 8π concentration of mass |

95

do this, we are going to prove the following two claims for each ε > 0 with parameters α = α(ε), η, and λ(ε) defined in (3.4.5). Claim 1. Suppose that ε and M satisfy the conditions 8π + ε(1 + η2 α/2) > M.

(3.4.11)

Then problem (3.2.1)–(3.2.3) has the property ℐε . Claim 2. Suppose that 8π + ε(1 + η2 α/2) ≤ M,

(3.4.12)

and problem (3.2.1)–(3.2.3) has the property ℐε(1+η2 α/2) . Then problem (3.2.1)–(3.2.3) has the property ℐε . Let us first prove that Claims 1 and 2 imply the property ℐε for all ε > 0. Obviously, inequality (3.4.11) holds true for ε = M − 8π. Thus, since α = α(ε) is a continuous function of ε, inequality (3.4.11) holds true for all ε ∈ (ε0 , M−8π] with some ε0 < M−8π, and problem (3.2.1)–(3.2.3) has the property ℐε in this range of ε by Claim 1. Recalling Remark 3.4.7, define ε0 = inf{ε > 0 : problem (3.2.1)–(3.2.3) has the property ℐε }, and suppose that ε0 > 0. By continuity, there exists ε1 > 0, such that ε1 (1 + η2 α(ε1 )/2) = ε0 . For every ε̃ ∈ (ε1 , ε0 ), we have the alternative: either ε̃ satisfies inequality (3.4.11) or inequality (3.4.12). In both cases, either by Claim 1 or Claim 2, problem (3.2.1)–(3.2.3) has the property ℐε̃ . This is a contradiction with the definition of ε0 because ε̃ < ε0 . Now, we prove both Claims 1 and 2 simultaneously, and the scheme of the proof is the following: If assumption (3.4.11) is satisfied, and if estimate (3.4.7) holds true for all t ∈ [0, Tb ), the proof of Claim 1 is completed by Lemma 3.4.6. At the first point t = T1 , where estimate (3.4.7) fails, we obtain inequality (3.4.12). Hence, using the recurrence hypothesis of Claim 2 and a suitable rescaling of the whole problem, we obtain Claim 2. Fix ε ∈ (0, M − 8π). Let η = η(ε), α, λ(ε) be defined by (3.4.5). Set γ2 =

αη2 ε , 2MB

(3.4.13)

and notice that γ 2 = γ 2 (ε) ≤ 1. Suppose that u0 satisfies conditions (3.4.10) with this value of γ. Thus, using inequality (3.4.2), we obtain w(0) ≥

∫ ψ(x)u0 (x) dx {|x|≤γ}

≥

󵄨 󵄨 ∫ u0 (x) dx − ∫ 󵄨󵄨󵄨1 − ψ(x)󵄨󵄨󵄨u0 (x) dx {|x|≤γ}

{|x|≤γ}

96 | 3 Blowups ≥ 8π + ε − B ∫ |x|2 u0 (x) dx {|x|≤γ}

≥ 8π + ε − Bγ 2 M.

(3.4.14)

Notice that, with our choice of γ in (3.4.13), we have MBγ 2 < ε/2. Thus, we obtain the inequality w(0) > 8π + ε/2, which is the first assumption (3.4.6) of Lemma 3.4.6. Next, we deal with the second assumption (3.4.7) of Lemma 3.4.6. Notice that if estimate (3.4.7) holds true for all t ∈ [0, Tb ). Then, by Lemma 3.4.6, we have the property ℐε with γ defined in (3.4.13). Suppose that estimate (3.4.7) does not hold for t = 0. Then, by assumption (3.4.10) and the inequalities γ < η and η2 ≤ 1/2, we obtain ∫ u0 (x) dx = {|x|≤λ}

∫ u0 (x) dx + {|x|≤γ}

u0 (x) dx

∫ {γ≤|x|≤λ}

≥ 8π + ε + εα ≥ 8π + ε(1 + η2 α/2).

Note that this inequality cannot be true under the condition (3.4.11) of Claim 1, because then the total mass of u0 would be greater than M. Thus, we have inequality (3.4.12) assumed in Claim 2. Suppose that the second assumption of Claim 2 is satisfied, namely, that each solution of problem (3.2.1)–(3.2.3) with an initial datum, satisfying (3.4.10) with ε replaced by ε(1 + η2 α/2) blows up at time estimated from above by λ2 T(M, ε(1 + η2 α/2)). Now, we rescale the solution, as explained in Remark 3.4.2, to see that it suffices choosing T(M, ε) = γ −2 λ2 T(M, ε(1 + η2 α/2)). Since γ, λ depend only on M, ε, we obtain the property ℐε . Now, consider the case when assumption (3.4.7) of Lemma 3.4.6 is not satisfied for some t ∈ (0, Tb ). Thus, by continuity, there exists T1 ∈ (0, Tb ) such that strict inequality (3.4.7) is satisfied for all t ∈ [0, T1 ), and for t = T1 , we have ∫

u(x, T1 ) dx = εα.

(3.4.15)

{η≤|x|≤λ}

Hence, by Lemma 3.4.6, the function w(t) is increasing for t ≤ T1 , and by (3.4.14), we obtain w(T1 ) = ∫ ψ(x)u(x, T1 ) dx ≥ w(0) ≥ 8π + ε − γ 2 MB. Now, the estimate 2 2

2

1{|x|≤λ} ≥ ψ(x) + (1 − (1 − η ) )1{η≤|x|≤λ} ≥ ψ(x) + η 1{η≤|x|≤λ}

implies ∫ u(x, T1 ) dx ≥ ∫ ψ(x)u(x, T1 ) dx + η2 {|x|≤λ}

∫ {η 0 is sufficiently large, that is, N ≥ cM with a constant c > 0 independent of M and of δ, then the solution u of problem (3.2.1)–(3.2.3), as long as this exists, satisfies for t > 0 ∫

u(x, t) dx ≥ ε exp(−Ct),

{δ/2≤|x|≤3R}

where C = C(M, ε, δ) > 0 depends only on M, δ, ε, and is independent of R. Proof. First, we assume that δ = 1 and 1 < R < ∞. Consider a weight function ϕ : ℝ+ → ℝ+ such that 0 { { { { (2s − 1)2 { { ϕ(s) = {1 { {(1 − s−R )2 { { 2R { {0

if 0 ≤ s ≤ 1/2, if 1/2 ≤ s ≤ 3/4, if 1 ≤ s ≤ R, if 2R ≤ s ≤ 3R, if 3R ≤ s.

Such a function ϕ can be chosen increasing on [0, R], decreasing on [R, 3R], supp ϕ ⊂ [1/2, 3R], and piecewise 𝒞 2 , with its derivatives ϕ(k) satisfying for k = 0, 1, 2: 󵄨󵄨 (k) 󵄨󵄨 C 󵄨󵄨ϕ (s)󵄨󵄨 ≤ k s

with a constant C independent of R and of s > 0.

(3.4.17)

We define, for the function Φ(x) = ϕ(|x|), the moment function of the solution u by H(t) = ∫ Φ(x)u(x, t) dx that measures mass of u contained in the annulus {1/2 ≤ |x| ≤ 3R}.

98 | 3 Blowups First, we present a particularly simple argument for radial solutions based on the identity from Lemma 2.1.8. For the evolution of H, we have the differential inequality d H(t) = ∫ Φ(x)Δu(x, t) dx + ∫ u(x, t)∇v(x, t) ⋅ ∇Φ(x) dx dt ϕ󸀠 (|x|) = ∫ ΔΦ(x) u(x, t) dx + ∫ u(x, t)(∇v(x, t) ⋅ x) dx |x| |ϕ󸀠 (|x|)| 1 M(|x|, t) ) dx ≥ ∫ u(x, t)(ΔΦ(x) − 2π |x| ≥ −C(M) ∫ Φ(x)u(x, t) dx = −C(M)H(t)

(3.4.18)

with a constant C(M) independent of R. In the last inequality, we have used the bound ΔΦ(x) −

1 󵄨 1 󵄨 M 󵄨󵄨ϕ󸀠 (|x|)󵄨󵄨󵄨 ≥ −C(M)Φ(x), 2π |x| 󵄨

which is valid since ΔΦ(x) ≥ C|x|−2 > 0 for 1/2 ≤ |x| ≤ 3/4 and 5/2R ≤ |x| ≤ 3R, and because of estimates (3.4.17). Since by the assumption, we have H(0) ≥ ε > 0, the above inequality yields the conclusion of Proposition 3.4.8 for radial solutions. Now, we prove Proposition 3.4.8 under the N-symmetry assumption. Our goal is to d derive again a differential inequality of the form dt H(t) ≥ −CH(t) with a suitably large constant C depending only on M and N, as was done in (3.4.18). Let us emphasize here that the crucial consequence of the N-symmetry assumption consists in some cancellations in the bilinear (with regard to u) integral ∫ u∇v ⋅ ∇Φ appearing in the first line of formula (3.4.18). Let us decompose the gradient of the weight function Φ as ∇Φ = μ + ν

(3.4.19)

1 supp μ ⊂ { ≤ |x| ≤ 2}, 2

(3.4.20)

supp ν ⊂ {R ≤ |x| ≤ 3R}.

(3.4.21)

with

and

We write

d H(t) dt

= I0 + I1 , where I0 = ∫ ΔΦ(x)u(x, t) dx and I1 = − ∬ u(x, t)u(y, t)

x−y ⋅ ∇Φ(x) dx dy ≡ I1,μ + I1,ν , |x − y|2

according to (3.4.20)–(3.4.21). Further, given A ≥ 4, we decompose the integral I1,μ into the sum of integrals I1,μ = J1 + J2 + J3

3.4 Blowup in the plane ℝ2 with 8π concentration of mass |

99

with the integration domains 1 1 1 ≤ |x| ≤ A, |y| ≥ A}, { ≤ |x| ≤ 2, |y| ≤ }, 4 4 4 1 1 { ≤ |x| ≤ A, ≤ |y| ≤ A}, 4 4 {

respectively. Note that, in fact, supp μ ⋐ {1/4 ≤ |x| ≤ 2}. The integrals J1 , J2 will be estimated rather crudely. Using the property (3.4.20), we have the following bound for the integral J1 : J1 = −

∫ u(x, t)u(y, t)

∫

{1/2≤|x|≤2} {|y|≥A}

≥− ≥−

2C M A

∫

x−y ⋅ μ(x) dx dy |x − y|2

u(x, t) dx

{1/2≤|x|≤2}

2CM ( A

∫

uΔΦ + C

{1/2≤|x|≤2}

uΦ),

∫ {1/2≤|x|≤2}

since for |x| ≤ 2 and 4 ≤ A ≤ |y| the relation |x − y| ≥ A/2 holds and, moreover, 1 ≤ ΔΦ(x) + CΦ(x) for each 1/2 ≤ |x| ≤ 2. Finally, let us define A = 4CM. Now, observe that for each ϱ > 0 there exists a constant Cϱ such that the both 1 “weights” |μ| and |μ| /2 are bounded by ΔΦ and Φ, that is, the inequality 󵄨1/2 󵄨 󵄨 󵄨󵄨 󵄨󵄨μ(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 ≤ ϱΔΦ(x) + Cϱ Φ(x)

(3.4.22)

holds. This inequality, together with |x − y| ≥ 1/4 and property (3.4.20), leads to the estimate J2 = −

∫

u(x, t)u(y, t)

∫

{1/2≤|x|≤2} {|y|≤1/4}

≥ −4ϱM

∫ {1/2≤|x|≤2}

x−y ⋅ μ(x) dx dy |x − y|2

ΔΦ(x)u(x, t) dx − Cϱ M

∫

Φ(x)u(x, t) dx,

{1/2≤|x|≤2}

1 , A = 6CM, and we obtain the required inequality. where we put ϱ = 24M To exploit some gain from the symmetry assumption (3.2.7) (that is, discover some cancellations in the integral J3 ), we decompose J3 further into the integrals J3,1 and J3,2 over the disjoint sets Ω1 and Ω2

Ω1 = {

1 1 ≤ |x| ≤ A, ≤ |y| ≤ A, x ∈ Γy } 4 4

Ω2 = {

1 1 ≤ |x| ≤ A, ≤ |y| ≤ A, x ∉ Γy }, 4 4

and

100 | 3 Blowups y x where Γy = {x : | |x| − |y| |≤

2π } is the sector determined by the direction of y. Obviously, N 4

we have x ∈ Γy ⇔ y ∈ Γx . Denote by S ⊂ ℝ the support of the function |μ(x)| + |μ(y)|. Since Ω1 is symmetric, and if (x, y) ∈ S, then either 1/2 < |x| < 2 or 1/2 < |y| < 2. So, for the first integral, we have the representation J3,1 = ∬ u(x, t)u(y, t) Ω1

=

x−y ⋅ μ(x) dx dy |x − y|2

(x − y) ⋅ (μ(x) − μ(y)) 1 dx dy ∬ u(x, t)u(y, t) 2 |x − y|2 Ω1

=

(x − y) ⋅ (∇Φ(x) − ∇Φ(y)) 1 dx dy. ∬ u(x, t)u(y, t) 2 |x − y|2 Ω1 ∩S

Therefore, we can estimate |J3,1 | by |J3,1 | ≤ 2

u(x, t)u(y, t) dx dy

∫ {1/4≤|x|≤A, 1/4≤|y|≤A, x∈Γy }∩S

≤C ≤C

4 M N

∫

u(x, t) dx,

{1/2≤|x|≤2}

4 M( N

∫

uΔΦ + C

{1/2≤|x|≤2}

since ∫

u(y, t) dy ≤ C

{1/4≤|y|≤2, y∈Γx }

∫

uΦ),

{1/2≤|x|≤2}

2 M N

for x ∈ supp μ,

due to the N-symmetry assumption. For the estimate of the integral J3,2 , note that the weight function μ satisfies |(x − y) ⋅ (μ(x) − μ(y))| |μ(x) − μ(y)| ≤ . |x − y| |x − y|2 Moreover, we have 󵄨 󵄨󵄨 󵄨󵄨μ(x) − μ(y)󵄨󵄨󵄨 ≤ C|x − y|,

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨μ(x) − μ(y)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨μ(y)󵄨󵄨󵄨.

So putting together those inequalities, we arrive at 1/2 󵄨 󵄨1/2 󵄨1/2 󵄨 󵄨 󵄨󵄨 󵄨󵄨μ(x) − μ(y)󵄨󵄨󵄨 ≤ C|x − y| (󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨μ(y)󵄨󵄨󵄨 ).

1 for some c > 0. Consequently, Observe that, if x ∉ Γy , |x|, |y| ≥ 1/2, then |x − y| ≥ cN using the symmetry of Ω2 , we obtain after splitting the integration domain into dyadic pieces

|J3,2 | ≤ C

∬ {1/2≤|x|≤A, 1/4≤|y|≤A, x∉Γy }

1

1

1 󵄨 /2 󵄨 /2 󵄨 󵄨 |x − y|− /2 (󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨μ(y)󵄨󵄨󵄨 )u(x, t)u(y, t) dx dy

3.4 Blowup in the plane ℝ2 with 8π concentration of mass | 101

≤C

−1/2

∑

(

1≤L≤cN,dyadic

×

L ) N

∫ {1/2≤|x|≤A, 1/4≤|y|≤A, x∉Γy , |x−y|≍L/N }

󵄨1/2 󵄨1/2 󵄨 󵄨 ∫(󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨μ(y)󵄨󵄨󵄨 )u(x, t)u(y, t) dx dy

with a constant C > 0. Observe that by the N-symmetry property, for 1/2 ≤ |x| ≤ 2, we have the bound u(y, t) dy ≤ C

∫ {|y|≥1/4, |x−y|≍L/N }

L M. N

Applying this, we estimate each summand by L 1 1 1 1 󵄨1/2 󵄨 󵄨1/2 󵄨 2CL− /2 N /2 ∫󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 u(x, t) dx M = CML /2 N − /2 ∫󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 u(x, t) dx. N 1

1

Since ∑1≤L≤cN,dyadic L /2 ≍ cN /2 , the entire sum is bounded from above by an application of inequality (3.4.22): 1

󵄨 /2 󵄨 CM ∫󵄨󵄨󵄨μ(x)󵄨󵄨󵄨 u(x, t) dx ≤ CϱM

ΔΦ(x)u(x, t) dx

∫ {1/2≤|x|≤2}

+ Cϱ M

Φ(x)u(x, t) dx.

∫ {1/2≤|x|≤2}

1 Now, we put ρ = 6CM , and adding the inequalities, we obtain the desired estimate 1 −|I1,μ | ≥ − 2 ∫{1/2≤|x|≤2} ΔΦ u − C ∫{1/2≤|x|≤2} Φ u. Similar considerations apply to the part containing the function ν in the decomposition (3.4.19), and we get

−|I1,ν | ≥ −

1 2

∫

ΔΦ u − CR−2

{R≤|x|≤3R}

∫

Φ u.

{R≤|x|≤3R}

We note that for R = 4, the proof is literally as the above, and the general case R ≥ 4 follows by scaling. Adding these inequalities, and using ΔΦ(x) = 0 for 2 ≤ |x| ≤ R, we obtain the differential inequality d H(t) ≥ −CH(t), dt similarly as was in (3.4.18). Note that we established that the assumption N ≥ CM, where C is large enough (but it does not depend on M), is sufficient in order that Proposition 3.4.8 holds for δ = 1. To complete the proof for arbitrary 0 < δ ≤ 1 < R, it suffices to use the rescaling defined in (3.2.8) with λ = 1/δ, and then to replace δR by R.

102 | 3 Blowups 3.4.3 Proofs of the 8π-mass concentration results Proof of Theorem 3.2.3. The rather natural idea of the proof is as follows: Suppose for contradiction that ∫{|x|≤R } u(x, tj ) dx ≥ 8π + ε for some sequences tj ↗ Tb , Rj ↘ 0, and j

a positive ε > 0. It suffices to construct a sequence {Sj } such that ∫

u(x, t) dx ≥ η(ε) > 0

for all t ∈ [tj , Tb ),

{Sj 0 is independent of j. If the annuli {x ∈ ℝ2 : Sj < |x| < Rj } are disjoint, then the total mass ∫ u(x, t) dx tends to infinity as t → Tb , which is impossible. Under the assumption that ∫{|x|≤R } u(x, tj ) dx ≥ 8π + ε for a positive ε > 0 and some j

sequences tj ↗ Tb , Rj ↘ 0, there exists a sufficiently large constant L depending only on M and ε, such that we have ∫ ψLRj (x)u(x, tj ) dx ≥ 8π + ε/2. To simplify the notation, we will denote from now on the radii LRj again by Rj . We also define a sequence 0 < Sj < Rj such that wSj (tj ) = 8π + ε/4, whereas, by our

choice, wRj (tj ) ≥ 8π + ε/2. Observe, that the rescaled function uj (x, t) = Sj2 u(Sj x, tj +Sj2 t) is a solution of the system (3.2.1)–(3.2.2) with the same mass M, see (3.2.8). Moreover, its initial condition uj (x, 0) = Sj2 u(Sj x, tj ) satisfies assumptions of Proposition 3.4.1, that is, ∫ uj (x, 0) dx ≥ ∫ ψ(x)uj (x, 0) dx ≥ 8π + {|x|≤1}

{|x|≤1}

ε . 4

Hence, the solution uj (x, t) cannot be classical after (by the definition, it means that this blows up not later than) Tw , where Tw = Tw (M, ε) can be chosen independently of j by Proposition 3.4.1. Denote by Tb (j) ≤ Tw (M, ε) the blowup time of the solution uj (x, t). By the definition of Sj and Rj and (3.3.2), we have wRj (tj ) − wSj (tj ) ≥

ε . 8

Passing to the rescaled solution uj (x, t), we obtain ∫(ψRj/Sj (x) − ψ(x))uj (x, 0) dx ≥

ε . 16

Since we have, by inequality (3.4.2), |ψRj/Sj (x) − ψ(x)| ≤ C|x|2 for some C = C(B) > 0, we obtain for an appropriate choice of the constant δ = δ(M, ε) (it suffices to choose δ2 (M, ε) = ε/(16CM)), still independent of j, ∫ {δ≤|x|≤Rj/Sj }

uj (x, 0) dx ≥

ε . 16

3.4 Blowup in the plane ℝ2 with 8π concentration of mass | 103

Applying Proposition 3.4.8, we infer that for t ≤ Tb (j), we have ∫ {δ/2≤|x|≤3Rj/Sj }

ε uj (x, t) dx ≥ β , 8

where the constant β > 0 depends only on Tw (M, ε), M, ε, and consequently ∫

uj (x, Tb (j)) dx ≥ β

{δ/2≤|x|≤3Rj/Sj }

ε 8

with the same constant β. Scaling back to the original coordinates, we get at the blowup time Tb ∫ {δ/2Sj ≤|x|≤3Rj }

ε u(x, Tb ) dx ≥ β . 8

(3.4.23)

So, if we choose Rj+1 (passing, if necessary, to a subsequence) satisfying 3Rj+1 ≤ δ/2Sj , so that the annuli {δ/2Sj ≤ |x| ≤ 3Rj } are disjoint, we infer that u(x, Tb ) accumulates infinite mass. Indeed, masses estimated in (3.4.23) (each bounded from below by the same positive number) are distributed over disjoint annuli {δ/2Sj ≤ |x| ≤ 3Rj }—a contradiction. Proof of Corollary 3.2.4. First, recall the main result obtained in Chapter 5, Section 5.3 (see [64]): if the initial condition satisfies ∫B u0 (x) dx ≤ 8π−ε for every ball B of radius δ, then the solution of problem (3.2.1)–(3.2.3) exists at least on the interval [0, T(M, δ, ε)], where the number T(M, δ, ε) > 0 depends on M, δ, ε, only. Now, let u be an N-symmetric solution with properties assumed in Theorem 3.2.3. Suppose that, for some fixed ε > 0 and δ > 0, there exists a sequence tj ↗ Tb such that ∫ u(x, tj ) dx ≤ 8π − ε

for each tj .

(3.4.24)

{|x|≤δ}

Using the N-symmetry property and (3.4.24), it is easy to show that there exists δ1 ∈ (0, δ] such that u(x, tj ) dx ≤ 8π − ε

∫

for each x0 ∈ ℝ2 and each tj

{|x−x0 |≤δ1 }

(use the N-symmetry if |x0 | > δ/2 and (3.4.24) if |x0 | < δ). Now, by [64], the solution of problem (3.2.1)–(3.2.3) with the initial condition u(x, tj ), Tb − tj < T(M, δ1 , ε)/2, exists at least on the interval [0, T(M, δ1 , ε)/2]. Thus, we have extended the solution beyond the blowup time Tb , which is a contradiction. Therefore, we have proved that lim inf ∫ u(x, t) dx ≥ 8π t→Tb

{|x| 0,

104 | 3 Blowups which, together with the upper bound in Theorem 3.2.3, completes the proof of this corollary.

3.5 Refinements of the proof of blowup of radial solutions in higher dimensions The blowup question for radially symmetric solutions of the d-dimensional Keller– d Segel model with d ≥ 3 in M /2 (ℝd ) has been recently studied in [50], and then those results have been improved, as below, in [51]. We begin with that case improving the result in the previous Sections ([50, Theorem 1.1]) using a refined argument from the proof of Theorem 3.2.1 in Section 3.2. Here, we formulate some new sufficient conditions for blowup of solutions of (3.2.1)–(3.2.2). Theorem 3.5.1 (Blowup of radial solutions with large concentration). If d ≥ 2, α = 2, u0 ∈ L1 (ℝd ) is a radially symmetric function and C x R2−d ∫ ψ( )u0 (x) dx > d σd , R d {|x|≤R}

for ψ(x) = (1 − |x|2 )2+ and some R > 0, then the solution u of the problem (3.2.1)–

(3.2.3) blows up in a finite time. Here, as usual, σd =

d

2π /2 Γ( d2 )

denotes the area of the unit

sphere in ℝd , see (11), Cd > 0 is a constant, which depends on the dimension only, and lim supd→∞ Cd ≤ 16. Thus, we show in the present Chapter that the radial concentration of data is the critical quantity for the finite time blowup of nonnegative radial solutions of (3.2.1)– (3.2.3). Here, we also use the ψ-radial concentration defined in (3.2.5) with a fixed radial nonnegative function ψ piecewise 𝒞 2 , supported on the unit ball such that ψ(0) = 1. Clearly, those quantities for such weight functions ψ are comparable, so we fix in the following ψ(x) = (1 − |x|2 )2+ , see (3.3.2) before. d Of course, for d ≥ 3 and p = d/2, the norm (17) in M /2 (ℝd ) (relevant also to the theory of existence of local-in-time solutions) dominates the radial concentration (19): |u0|M d/2 ≥ R2−d ∫ u0 (x) dx. {|x|≤R}

But, in fact, for radially symmetric u0 , the quantities |u0|M d/2 and ||u0 ||ψ are equivalent, see Proposition 2.7.1 and Remark 3.5.2. This “criticality” implies that for the initial data with small ||u0 ||ψ solutions exist indefinitely in time (see [16], [161]), whereas the result of Theorem 3.5.1 shows that for sufficiently large ||u0 ||ψ regular solutions cease to exist in a finite time. This result

3.5 Refinements of the proof of blowup of radial solutions in higher dimensions | 105

seems new for d ≥ 3, although related criteria, formulated in terms of global quantities, such as the second moment ∫ |x|2 u0 (x) dx, appeared in e. g. [39]. In fact, this astonishingly simple proof, involving an analysis of a local moment of the solution, extends to the two dimensional case x ∈ ℝ2 of arbitrary (not necessarily radially symmetric) nonnegative solutions, see [194], [163] for a similar argument with different weight functions, and Theorem 3.6.1 in the following Section 3.6. Proof of Theorem 3.5.1. The idea is to apply an optimization, which improves the rough estimate in Section 3.3 and [50, 51]. We will derive a differential inequality for the local moment of the solution wR (t) = ∫ ψR (x)u(x, t) dx

(3.5.1)

again with the scaled weight function ψR in (3.3.2) supported on the ball {|x| ≤ R}. The function ψ ∈ 𝒞 1 (ℝd ) has piecewise continuous and bounded second derivatives, see (3.3.3). Now, using equation (3.2.1), integrations by parts, and applying relations (3.3.2)– (3.3.3), we obtain d w (t) = ∫ ΔψR (x)u(x, t) dx + ∫ u(x, t)∇v(x, t) ⋅ ∇ψR (x) dx dt R |x|2 = R−2 ( ∫ (−4d + 4(d + 2) 2 )u(x, t) dx R {|x|≤R}

− 4 ∫ u(x, t)(∇v(x, t) ⋅ x)(1 − {|x|≤R}

|x|2 ) dx). R2

(3.5.2)

Thus, using the assumption on radial symmetry of u, the useful relation 1 1−d 󸀠 R M (R) σd

u(x) =

for |x| = R,

and Lemma 2.1.8, we get 1

1 2 d R w (t) = R ∫ M 󸀠 (Rr, t)(−d + (d + 2)r 2 ) dr 4 dt R 0

1

R2−d + R ∫ M 󸀠 (Rr, t)M(Rr, t)r 2−d (1 − r 2 ) dr. σd 0

Notice also the following (obvious) relations: 1

M(R, t) =

∫ u(y, t) dy = R ∫ M 󸀠 (Rr, t) dr {|y|≤R}

0

(3.5.3)

106 | 3 Blowups 1

2

≥ wR (t) = R ∫ M 󸀠 (Rr, t)(1 − r 2 ) dr 0

1

= 4 ∫ M(Rr, t)r(1 − r 2 ) dr.

(3.5.4)

0

Now, let λ ∈ ℝ be a number to be fixed later on. Since 1

wR (t) = 4R ∫ M(Rr, t)r(1 − r 2 ) dr, 0

we may rewrite equation (3.5.3) after the integration by parts and use (3.5.4) as follows: 1

1

1 2 d R w (t) = 2 ∫ M 󸀠 (Rr, t)r 2 dr − 2d ∫ M(Rr, t)r dr 4 dt R 0

0

1

− λ(4 ∫ M(Rr, t)r(1 − r 2 ) dr − wR (t)) 0

+

1

R2−d ∫ M(Rr, t)2 r 1−d ((d − 2) − (d − 4)r 2 ) dr 2σd 0

≥ λwR (t)

1

R2−d + ∫((d − 2) − (d − 4)r 2 ) 2σd 0

× (M(Rr, t)r 1

1−d 2

2

−

1+d 2σd d + 2λ(1 − r 2 ) r 2 ) dr 2 2−d R (d − 2) − (d − 4)r

2σd (d + 2λ(1 − r 2 ))2 d+1 − 2−d r dr. ∫ (d − 2) − (d − 4)r 2 R 0

Thus, we may estimate the right-hand side as 1

2σd 1 2 d (d + 2λ(1 − r 2 ))2 d+1 r dr. R wR (t) ≥ λwR (t) − 2−d ∫ 4 dt (d − 2) − (d − 4)r 2 R 0

By direct calculations, the function 1

1 (d + 2λ(1 − r 2 ))2 d+1 F(λ) ≡ ∫ r dr λ (d − 2) − (d − 4)r 2 0

(3.5.5)

3.5 Refinements of the proof of blowup of radial solutions in higher dimensions | 107

satisfies 1

C min F(λ) = F(√ ) = 2√AC + B, λ>0 A 1

B = 4d ∫ 0

where A = 4 ∫ 0

(1 − r 2 ) r d+1 dr, (d − 2) − (d − 4)r 2

1

C = d2 ∫ 0

(1 − r 2 )2 r d+1 dr, (d − 2) − (d − 4)r 2

r d+1 dr. (d − 2) − (d − 4)r 2

Thus, it is clear from (3.5.5) that if R2−d wR (0) > 2σd (2√AC + B),

(3.5.6)

d then 41 R2 dt wR (t) ≥ δ for some δ > 0 since wR (t) increases. As a consequence, the function wR (t) becomes greater than M = ∫ u(x, t) dx in a finite time, which is a contradiction with the existence of nonnegative, mass-conserving solutions. Finally, let us express condition (3.5.6) in a more explicit way. By the Cauchy– Schwarz inequality, we see that B ≤ 2√AC. Evidently, for d ≥ 3, we have 1

4 8 , A≤ ∫(1 − r 2 )r d+1 dr = d−2 (d − 2)(d + 2)(d + 4) 0

and C≤d

21

2

1

∫ r d+1 dr = 0

d2 . 2(d + 2)

For an estimate of the constant Cd for d ≥ 3, note that (3.5.6) is satisfied if, for example, R2−d wR (0) > 16

1 d √ σ . d + 2 (d − 2)(d + 4) d

Thus, the condition (3.5.7) follows for large d if ||u0 ||ψ > conditions ||u0 ||ψ >

Cd σ d d

and R

2−d

wR (0) >

Cd σ d d

Cd σ d d

(3.5.7)

with a Cd = 16, since the

for a particular R > 0 are equivalent.

In the following remark, we give a comparison of the moment wR with the radial distribution function evaluated at some other argument R√1 − s. Remark 3.5.2. Note that for nonnegative locally integrable functions ω, each R > 0, and for each s ∈ (0, 1) we have ∫ ψR (x)ω(x) dx ≥

∫ |x|≤R√1−s

2

(1 − (1 − s)) ω(x) dx = s2

∫ |x|≤R√1−s

ω(x) dx,

108 | 3 Blowups and 2

d/2−1

max s (1 − s)

s∈[0,1]

2

d/2−1

d−2 4 )( ) =( d+2 d+2

≡ Hd

so that ||ω|| ψ ≥ Hd L if |ω||M d/2 > L, because (R√1 − s)2−d

∫

ω(x) dx > L

|x|≤R√1−s

holds for some R > 0 and s =

4 . d+2

Therefore,

|u0|M d/2 >

Cd σ H −1 d d d

is a sufficient condition for the occurrence of the blowup for radial initial data u0 ≥ 0 expressed in terms of the Morrey norms. Note that lim sup( d→∞

Cd σ H −1 )/(e2 dσd ) ≤ 1 d d d

as d → ∞,

(3.5.8)

so |u0|M d/2 > e2 dσd is a simpler sufficient condition for blowup. Remark 3.5.3. One may improve the conditions (3.5.6) and (3.5.7) for blowup (that is, diminish the numerical value of Cd ) by solving the following rescaled variational problem: inf ℱ [φ],

where 1

2

1

ℱ [φ] = 2 ∫ φ (r)r dr − 2d ∫ φ(r)r dr 󸀠

0

+

0

1

1 ∫ φ(r)2 r 1−d ((d − 2) − (d − 4)r 2 ) dr 2 0

with the constraints 1

φ(0) = 0,

̃ φ(1) = A,

̃ ∫ φ(r)r(1 − r 2 ) dr = B,

φ󸀠 (r) ≥ 0.

(3.5.9)

0

Indeed, by relations (3.5.3)–(3.5.4), we have 1 2 d R w (t) ≥ σd Rd−2 inf ℱ [φ], 4 dt R ̃ = M and where φ is chosen in such a way to have relations (3.5.9), namely, Rd−2 σd A d−2 ̃ R σd B = wR (t). The monotonicity of φ reflects the increasing property of M(⋅, t), and ̃ ∈ ( 1 A, ̃ 1 A) ̃ is here an obvious restriction. B 4 6√3

3.6 Blowup of solutions for the two dimensional fractional diffusion Keller–Segel model | 109

To solve the variational problem (3.5.9)–(3.5.9), let Eλ (r) =

d+2λ(1−r 2 ) d r with (d−2)−(d−4)r 2 󸀠

a

parameter λ ∈ ℝ. Whenever Eλ is not an increasing function, the constraint φ (r) ≥ 0 leads to the consideration of “broken” extremals, that is, 0 { { φλ (r) = {Eλ (r − r1 ) {̃ {A

if 0 ≤ r ≤ r1 , if r1 < r < r2 , if r2 ≤ r ≤ 1,

̃ = Eλ (r2 − r1 ), and 0 ≤ r1 < r2 ≤ 1 are chosen so where Eλ is nondecreasing on [r1 , r2 ], A 1 2 ̃ that ∫0 φλ (r)r(1−r ) dr = B. One can show that the functional ℱ [φ] attains its minimum for those φλ . Remark 3.5.4. Another improvement of the sufficient condition (3.5.8) for blowup (up to twice better asymptotic bound) can be obtained along the computations in (3.5.3), (3.5.4) with the use of inequality (3.3.4). First, we write 1

1

0

0

wR2 (t) ≤ 16 ∫ M 2 r 1−d (1 − r 2 ) dr × ∫ r 1+d (1 − r 2 ) dr, so that (3.5.3) entails 1 2 d (d + 2)2 R wR (t) ≥ − wR (t) 4 dt 2 R2−d (d − 2)(d + 2)(d + 4) + wR2 (t)2 σd 32 =

R2−d (d − 2)(d + 4) d+2 wR (t)(−(d + 2) + wR (t)). 2 σd 8

Now, if R2−d wR (0) > 8σd

d+2 , (d − 2)(d + 4)

then wR (t) strictly increases, and becomes wR (t) > M in a finite time, a contradiction. C Therefore, |u0|M d/2 > dd σd Hd−1 ≍ 21 e2 dσd leads to a blowup.

3.6 Blowup of solutions for the two dimensional fractional diffusion Keller–Segel model Now, we consider the system with fractional diffusion operator in two space dimensions α

ut + (−Δ) /2 u + ∇ ⋅ (u∇v) = 0, Δv + u = 0,

x ∈ ℝ2 , t > 0, 2

x ∈ ℝ , t > 0,

(3.6.1) (3.6.2)

110 | 3 Blowups supplemented with the nonnegative initial condition u(x, 0) = u0 (x).

(3.6.3)

Many previous works have dealt with the existence of global-in-time solutions with small data in critical Morrey spaces, that is, those which are scale invariant under a natural scaling of the chemotaxis model, see [16] and [172]. Our criteria for a blowup of solutions with large concentration in [28] can be expressed using Morrey 2 space norms M /α (ℝ2 ), see Remark 3.6.2 for more detail, and the size of such a norm is critical for the global-in-time existence versus finite time blowup. Theorem 3.6.1 (Blowup in two dimensional models with fractional diffusion). Consider a local-in-time nonnegative solution of problem (3.2.1)–(3.2.3) with a nonnegative function u0 on ℝ2 and α ∈ (0, 2). i) (Blowup for general solutions of the two dimensional Keller–Segel model with fractional diffusion). If there exist x0 ∈ ℝ2 and R > 0 such that Rα−2

u0 (y) dy > Cα

∫

and

{|y−x0 |≤R}

∫

u0 (y) dy < ν,

(3.6.4)

{|y−x0 |>R}

for some explicit constants: small ν > 0 and large Cα > 0, then the solution u blows up in a finite time. ii) (Radial blowup for the Keller–Segel model with fractional diffusion). If α ∈ [1, 2), ̃ such that for each radially symmetric initial data u satisfying then there exists C α 0 ̃ Rα−2 ∫ u0 (y) dy > C α

(3.6.5)

|y|≤R

for some R > 0, the solution u of problem (3.2.1)–(3.2.3) blows up in a finite time. Remark 3.6.2. The cases i) and ii): α < 2. The first condition in (3.6.4) is equivalent 2 to a sufficiently large Morrey norm of u0 in the space M /α (ℝ2 ). Indeed, we have the obvious relations |u0|M 2/α ≥ Rα−2

∫

u0 (y) dy

{|y−x0 |≤R}

for every x0 and R > 0, but also there is x0 ∈ ℝ2 and R > 0 such that |u0|M 2/α ≤ 2Rα−2

∫

u0 (y) dy.

{|y−x0 |≤R}

Thus, our blowup condition in terms of the Morrey norm M /α (ℝ2 ) seems to be new, and in a sense complementary to that guaranteeing the global-in-time existence of 2

3.6 Blowup of solutions for the two dimensional fractional diffusion Keller–Segel model | 111

solutions, where smallness of initial conditions in the M /α (ℝ2 ) Morrey norm has to be imposed, see prototypes of such results in [16, Theorem 1], [41, Remark 2.7] and [172, Theorem 2]. Similarly, (3.6.5) implies that the Morrey norm |u0|M 2/α is large. 2

Remember that the natural scaling for system (3.6.1)–(3.6.2) uλ (x, t) = λα u(λx, λα t) leads to the equality ∫ uλ dx = λα−2 ∫ u dx. In particular, when α ∈ (0, 2), mass of rescaled solution uλ can be chosen arbitrarily with a suitable λ > 0. Thus, the first part of the condition in Theorem 3.6.1(i) is insensitive to the actual value of M, so without loss of generalization we may suppose that M = 1. The second part of the condition (3.6.4) is not scale invariant. However, we believe that this assumption is not necessary for the conclusion in Theorem 3.6.1(i). In fact, one can prove it for α close to 2 by an inspection of methods from [16, 50]. Remark 3.6.3. In view of results in Chapter 5, Section 5.3, see also [64], assumptions on u0 in Theorem 3.6.1 can be relaxed; it suffices that u0 is a finite nonnegative measure satisfying some “small atoms” assumption. We prove Theorem 3.6.1 using the method of truncated moments, which is reminiscent of that in the papers [194, 163]. The “bump” function ψ defined in (3.3.2) satisfies (see (3.3.3)) Δψ(x) = (−8 + 16|x|2 ) ≥ −8ψ(x) ≥ −8

for

|x| < 1,

(3.6.6)

and ψ is strictly concave in a neighbourhood of x = 0. For the readers’ convenience, we recall now auxiliary lemmata from [28]. Lemma 3.6.4. For each ε ∈ (0, √13 ), the function ψ defined in (3.3.2) is strictly concave for all |x| ≤ ε. More precisely, ψ satisfies Hψ ≤ −θ(ε)I

(3.6.7)

for all |x| ≤ ε, where Hψ is the Hessian matrix of second derivatives of ψ, θ(ε) = 4(1−3ε2 ), and I is the identity matrix. In particular, we have θ(ε) ↗ 4

as

ε ↘ 0.

(3.6.8)

Proof. For every ξ ∈ ℝ2 , the following identity holds ξ ⋅ Hψ ξ = 4(−|ξ |2 (1 − |x|2 ) + 2(x ⋅ ξ )2 ). Thus, by the Cauchy–Schwarz inequality, we have ξ ⋅ Hψ ξ ≤ 4|ξ |2 (3|x|2 − 1). Next, we recall a well known property of concave functions, and we formulate a crucial inequality in our proof of the blowup result.

112 | 3 Blowups Lemma 3.6.5. For every function Ψ : ℝ2 → ℝ, which is strictly concave on a convex domain Ω ⊂ ℝ2 , we have for all x, y ∈ Ω (x − y) ⋅ (∇Ψ(x) − ∇Ψ(y)) ≤ −θ|x − y|2 ,

(3.6.9)

where θ > 0 is the constant of strict concavity of Ψ on Ω, that is, satisfying HΨ ≤ −θ I. In particular, for the fundamental solution E2 (x) of the Laplacian on ℝ2 , which sat1 x isfies ∇E2 (x) = − 2π , and a strictly concave function Ψ, we have for all x, y on the |x|2 domain of the strict concavity of Ψ ∇E2 (x − y) ⋅ (∇Ψ(x) − ∇Ψ(y)) ≥

θ . 2π

(3.6.10)

Proof. By the concavity, we obtain Ψ(x) ≤ Ψ(y) + ∇Ψ(y) ⋅ (x − y) −

θ |x − y|2 . 2!

Summing this inequality with its symmetrized version (with x, y interchanged) leads to the claim. We have the following scaling property of the fractional Laplacian: α

α

(−Δ) /2 ψR (x) = R−α ((−Δ) /2 ψ)R ,

(3.6.11) α

and we notice the following boundedness property of (−Δ) /2 ψ: Lemma 3.6.6. For every α ∈ (0, 2], there exists a constant kα > 0 such that α/2 󵄨 󵄨󵄨 󵄨󵄨(−Δ) ψ(x)󵄨󵄨󵄨 ≤ kα .

(3.6.12)

α

In particular, k2 = 8 by (3.3.4). Moreover, (−Δ) /2 ψ(x) ≤ 0 for |x| ≥ 1. Proof. For α = 2, this is an obvious consequence of the explicit form of the weight ψ, hence we assume α ∈ (0, 2). To show estimate (3.6.12) for α ∈ (0, 2), it suffices to use the representation of the fractional Laplacian with α ∈ (0, 2) in (23). Now, using the Taylor formula together with the fact that ψ, D2 ψ ∈ L∞ (ℝ2 ), we immediately obtain that the integral on the right-hand side is finite and uniformly bounded in x ∈ ℝ2 . Since ψ(x) ≥ 0, ψ ≢ 0, and ψ(x) = 0 for |x| ≥ 1, we have for all |x| ≥ 1 α

(−Δ) /2 ψ(x) = −𝒜 P.V. ∫

ψ(x + y) dy < 0. |y|2+α

(3.6.13)

Now, we prove our second blowup result for not necessarily radial solutions.

3.6 Blowup of solutions for the two dimensional fractional diffusion Keller–Segel model | 113

Proof of Theorem 3.6.1. We consider again the quantity wR , defined in (3.5.1). Let MR (t) ≡

∫ u(x, t) dx ≥ wR (t)

(3.6.14)

{|x|≤R}

denote mass of the distribution u contained in the ball {|x| ≤ R} at time t. Now, using equation (3.6.1), we determine the evolution of wR (t) d α w (t) = − ∫(−Δ) /2 u(x, t)ψR (x) dx + ∫ u(x, t)∇v(x, t) ⋅ ∇ψR (x) dx dt R α

= − ∫ u(x, t)(−Δ) /2 ψR (x) dx +

(3.6.15)

1 ∬ u(x, t)u(y, t)∇E2 (x − y) ⋅ (∇ψR (x) − ∇ψR (y)) dy dx, 2

where we applied the formula v = E2 ∗ u, and the last expression follows by the symmetrization of the double integral: x 󳨃→ y, y 󳨃→ x. Since u(x, t) ≥ 0, by the scaling relation (3.6.11) and Lemma 3.6.6, we obtain α

− ∫ u(x, t)(−Δ) /2 ψR (x) dx ≥ −R−α kα ∫ u(x, t) dx.

(3.6.16)

{|x|≤R}

Now, let ε ∈ (0, √13 ). By Lemma 3.6.4, the weight function ψR in (3.3.2) is concave for |x| ≤ εR with a concavity constant θ = R−2 θ(ε). Thus, by Lemma 3.6.5, we have ∇E2 (x − y) ⋅ (∇ψR (x) − ∇ψR (y)) ≥ R−2

θ(ε) 2π

for |x|, |y| < εR. Hence, the bilinear term on the right-hand side of (3.6.15) satisfies 1 ∬ u(x, t)u(y, t)∇E2 (x − y) ⋅ (∇ψR (x) − ∇ψR (y)) dy dx 2 θ(ε) 1 ≥R−2 ∫ ∫ u(x, t)u(y, t) dy dx + J, 4π 2

(3.6.17)

{|x|≤εR} {|y|≤εR}

where the letter J denotes the integral J=

u(x, t)u(y, t)∇E2 (x − y) ⋅ (∇ψR (x) − ∇ψR (y)) dy dx.

∬ ℝ2 ×ℝ2 \({|x| 0 small enough, we get blowup in the optimal range M > 8π. Indeed, if M > 8π, then there exists ε > 0 small, and R ≥ R(ε) > 0 sufficiently large so that wR (0) is sufficiently close to M, and we have −8 +

1 w (0) + C(ε)(wR (0) − M) > 0. π R

Case ii) In the case α < 2, the blowup occurs if for some R > 0 the quantity Rα−2 ∫ u0 (x) dx {|x|≤R}

3.7 Keller–Segel model with chemoattractant consumption terms | 115

is large enough, and simultaneously u0 is well concentrated, that is, C(ε)(M − wR (0)) is small. Case iii) First, observe that for each α ∈ (0, 2] there exists ℓα > 0 such that α

(−Δ) /2 ψ ≤ ℓα ψ. α

Indeed, this is a consequence of the fact that sup|x|=1 (−Δ) /2 ψ(x) < 0 (see equation (3), α and notice that ψ ≢ 0) and the continuity of (−Δ) /2 ψ on ℝ2 . Note also that ℓ2 = 8 is the optimal constant. We infer from the computations in (3.3.5)–(3.5.3) that 1

d R−2 wR (t) ≥ −ℓα R−α wR (t) + 4 ∫ M(Rr, t)2 r dr dt 2π 0

3R w (t)2 4π R −2

≥ −ℓα R−α wR (t) +

(3.6.21)

since by (3.5.4) we have 1

2

2

1

2

1

2

(4 ∫ M(Rr, t)r(1 − r ) dr) ≤ 16 ∫ M(Rr, t) r dr ∫ r(1 − r 2 ) dr 0

0 1

=

0

8 ∫ M(Rr, t)2 r dr. 3 0

Therefore, if Rα−2 wR (0) >

4π ℓ , 3 α

then we have d w (t) ≥ δ > 0 dt R

for some δ. We arrive at a contradiction with existence of global nonnegative solutions as was in the proof of Theorem 3.5.1. It is worth noting that the blowup time obtained from the argument in (3.6.21) depends only on R and wR (0), unlike in case (ii), of not necessarily radial solutions.

3.7 Keller–Segel model with chemoattractant consumption terms Results concerning the system (3.2.1)–(3.2.2) in previous sections are generalized (as in [28]) to the case of the parabolic-elliptic Keller–Segel model with the consumption of the chemoattractant effect, that is, with (3.2.2) replaced by the equation Δv −γv +u = 0, that is, (3.7.2). In such a case, the Newtonian potential E2 is replaced by the Bessel kernel Kγ of the operator (−Δ + γ)−1 on ℝ2 satisfying the relation ∇Kγ (x) = −

1 x g (|x|) 2π |x|2 γ

116 | 3 Blowups 1/2

with a decreasing smooth function gγ such that gγ (0) = 1, and gγ (|x|) ≤ Ce−γ |x| . For more general kernels, describing the spreading of the chemoattractant, and leading to blowup of solutions, see also [151, 12, 42, 13, 170] and Chapter 5, Section 5.4. Thus, we consider in this section the following version of the parabolic-elliptic Keller–Segel model of chemotaxis in two space dimensions: α

ut + (−Δ) /2 u + ∇ ⋅ (u∇v) = 0, Δv − γv + u = 0,

x ∈ ℝ2 , t > 0, 2

x ∈ ℝ , t > 0,

(3.7.1) (3.7.2)

supplemented with the initial condition u(x, 0) = u0 (x).

(3.7.3)

Here, the unknown variables u = u(x, t) and v = v(x, t) denote the density of the population, and the density of the chemical secreted by the microorganisms, respectively, and the given consumption (or degradation) rate of the chemical is denoted by γ ≥ 0 in the Helmholtz equation (3.7.2). The diffusion operator is described either by the usual α Laplacian (α = 2) or by a fractional power of the Laplacian (−Δ) /2 with α ∈ (0, 2). The initial data are nonnegative functions u0 ∈ L1 (ℝ2 ) of the total mass M = ∫ u0 (x) dx. Our main results include criteria for blowup of nonnegative solutions of problem (3.7.1)–(3.7.3) expressed in terms of a local concentration of data (Theorem 3.7.1), and the existence of global-in-time solutions for the initial condition of an arbitrary mass M, and each sufficiently large γ (Theorem 3.7.5). The novelty of these blowup results consists in using local properties of solutions instead of a comparison of the total mass and moments of a solution as was done in [194], [160], [176, 177, 178], [62], [42], [151], and [41]. In particular, we complement the result in [160] by saying that solutions of (3.7.1)–(3.7.3) with α = 2, fixed γ ≥ 0 and sufficiently well-concentrated u0 with M > 8π blow up in a finite time, by showing that solutions of that system with u0 of arbitrary M > 0 and all sufficiently large γ are global-in-time. Many previous works have dealt with the existence of global-in-time solutions with small data in critical Morrey spaces, that is, those which are scale invariant under a natural scaling of the chemotaxis model, see [16] and [172]. Our criteria for a blowup of solutions with large concentration can be expressed again in terms of Morrey space norms (see Remark 3.7.3 for more detail), and we have found that the size of such a norm is critical for the global-in-time existence versus finite time blowup.

3.7.1 Statement of results for the system with consumption It is well known that problem (3.7.1)–(3.7.3) with α = 2 has a unique mild solution u ∈ 𝒞 ([0, T); L1 (ℝ2 )) for every u0 ∈ L1 (ℝ2 ) and γ ≥ 0. Here, as usual, a mild solution satisfies a suitable integral formulation of the Cauchy problem (3.7.1)–(3.7.3). Due to a

3.7 Keller–Segel model with chemoattractant consumption terms | 117

parabolic regularization effect (following, for example, [131, Theorem 4.2]), this solution is smooth for t > 0, hence, it satisfies the Cauchy problem in the classical sense. Moreover, it conserves the total mass ∫ u(x, t) dx = ∫ u0 (x) dx = M

for all

t ∈ [0, T),

ℝ2

ℝ2

and is nonnegative when u0 ≥ 0. Proofs of these classical results can be found in e. g. [160, 172, 163, 151, 69], see also Chapter 3, Section 3.5. Analogous results on local-intime solutions to the Cauchy problem (3.7.1)–(3.7.3) with α ∈ (1, 2) have been obtained in [41], [172, Theorem 2]. To the best of our knowledge, [178, Theorem 1.1] and a recent [230, Theorem 1, Theorem 2] are the only results on local-in-time classical solutions of the Cauchy problem (3.7.1)–(3.7.3) with α ∈ (0, 1], d ≥ 2. Thus, the case ii) of Theorem 3.7.1 asserts that such a solution cannot be global-in-time for initial data satisfying (3.7.5) below. In our first result, we formulate new sufficient conditions for blowup (that is, nonexistence for all t > 0) of such local-in-time solutions of problem (3.7.1)–(3.7.2). Theorem 3.7.1. Consider u ∈ 𝒞 ([0, T); L1 (ℝ2 ))—a local-in-time nonnegative classical solution of problem (3.7.1)–(3.7.3) with a nonnegative u0 ∈ L1 (ℝ2 ). i) Let α = 2 and γ > 0 (the Keller–Segel model with the consumption). If M > 8π, and if u0 is well concentrated around a point x0 ∈ ℝ2 , namely, there exists R > 0 such that e−√γR

∫

u0 (y) dy > 8π

and

{|y−x0 | 0, then the solution u blows up in a finite time. ii) Let α ∈ (0, 2) and γ ≥ 0 (the Keller–Segel model with fractional diffusion). If there exist x0 ∈ ℝ2 and R > 0 such that Rα−2

∫ {|y−x0 | C

and

∫

u0 (y) dy < ν,

(3.7.5)

{|y−x0 |≥R}

for some explicit constants: small ν > 0 and big C > 0 depending on γ, then the solution u ceases to exists in a finite time. Remark 3.7.2. Case (i) α = 2 and γ > 0 has been considered in [160], but the sufficient conditions for blowup were expressed in terms of globally defined quantities: that is, mass M > 8π and the moment ∫ |x|2 u0 (x) dx < ∞.

118 | 3 Blowups Remark 3.7.3. Case (ii) α < 2. The first condition in (3.7.5) is equivalent to a sufficiently 2 large Morrey norm of u0 in the space M /α (ℝ2 ). Indeed, by (17), obviously, we have |u||M p =

sup

R>0, x∈ℝ2

R2( /p−1) 1

󵄨 󵄨󵄨 󵄨󵄨u(y)󵄨󵄨󵄨 dy < ∞

∫ {|y−x| 0 such that |u0|M 2/α ≤ 2Rα−2

∫

u0 (y) dy.

{|y−x0 | 0. Thus, the conditions in Theorem 3.7.1(ii) are insensitive to the actual value of M, so without loss of generality we may suppose that M = 1. The second parts of the condition (3.7.4) and (3.7.5) are not scaling invariant. However, we believe that these assumptions are not necessary for the conclusion in Theorem 3.7.1(i, ii). In fact, one can prove it for α close to 2 by an inspection of methods in [50, 64]. Next, we show that the first condition in the concentration assumptions (3.7.4) is in some sense optimal to obtain a blowup of solutions. In the following theorem, we show that for every initial integrable function u0 , even with its L1 norm above 8π, the corresponding mild solution to the model (3.7.1)–(3.7.3) with α = 2 is global-in-time for all sufficiently large consumption rates γ > 0. This result appears in the present chapter even if the formal place for this would be in Chapter 2 on global-in-time solutions, because of the situation of the topic: Keller– Segel model with chemoattractant consumption, its blowup properties versus time global existence. Theorem 3.7.5. Let α = 2, γ > 0. For each u0 ∈ L1 (ℝ2 ), there exists γ(u0 ) > 0 such that for all γ ≥ γ(u0 ) the Cauchy problem (3.7.1)–(3.7.3) has a global-in-time mild solution

3.7 Keller–Segel model with chemoattractant consumption terms | 119

satisfying u ∈ 𝒞 ([0, ∞); L1 (ℝ2 )). This is a classical solution of system (3.7.1)–(3.7.2) for t > 0, and satisfies for each p ∈ [1, ∞) the decay estimates 󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p < ∞. t>0

(3.7.6)

Thus, for each u0 (not necessarily nonnegative) and γ large enough, depending on u0 ∈ L1 (ℝ2 ), solutions of the Cauchy problem are global-in-time, so there is no critical value of mass, which leads to a blowup of solutions. On the other hand, if M > 8π, then for 0 ≤ γ ≪ 1 the solutions blow up in a finite time, as it is seen from the sufficient conditions for blowup in Theorem 3.7.1(ii). Lemma 3.7.6. For every γ > 0, the operator (−Δ + γ)−1 solving the Helmholtz equation (3.7.2) satisfies 󵄩󵄩 1/p−1/q−1/2 −1 󵄩 ‖z‖p , 󵄩󵄩∇(−Δ + γ) z 󵄩󵄩󵄩q ≤ Cγ for every 1 ≤ p < 2 < q < ∞ such that

critical case

1 p

−

1 q

= 21 −1

kernel Kγ of (−Δ + γ)

1 p

−

1 q


1. Moreover, the Bessel

has the following pointwise behavior at 0 and ∞ 1 x as x → 0, 2π |x|2 1 −√γ|x| 󵄨 󵄨󵄨 as x → ∞, 󵄨󵄨∇Kγ (x)󵄨󵄨󵄨 ≤ C e |x| ∇Kγ (x) ≈ −

(3.7.8) (3.7.9)

and satisfies the global one-sided bound x ⋅ ∇Kγ (x) ≤ −

1 −√γ|x| . e 2π

(3.7.10)

Proof. The proof of inequality (3.7.7) requires separate arguments in two cases, p1 − q1 < 1 2

and

1 p

−

1 q

= 21 . In the first case, the result is a consequence of inequalities (39) and

(40) by representing the operator (−Δ + γ)−1 as the Laplace transform of heat kernels ∞

(−Δ + γ)−1 = ∫ e−γs esΔ ds. 0

Indeed, we have the following representation of Kγ in the Fourier variables: ? (K γ ∗ z)(ξ ) =

∞

2 1 ̂ ) = ∫ e−γs e−s|ξ | z(ξ ̂ ) ds, z(ξ 2 |ξ | + γ

0

so that ∞

󵄩󵄩 −1 󵄩 −γs 1/q−1/p−1/2 ds ‖z‖p 󵄩󵄩∇(−Δ + γ) z 󵄩󵄩󵄩q ≤ C ∫ e s 0

(3.7.11)

120 | 3 Blowups ∞

≤ Cγ 1/p−1/q− /2 ∫ e−s s− /2+1/q−1/p ds ‖z‖p , 1

1

0

the latter integral is finite due to the assumption on p and q. When p1 − q1 = 21 , inequality (3.7.7) follows from the end-point case of the Sobolev

inequality ‖∇(−Δ)−1 u‖q ≤ C‖u‖p . For properties (3.7.8), (3.7.9), and (3.7.10), see [160, Lemma 3.1] and [226, Chapter V, Section 6.5].

Remark 3.7.7. Let us note that the reference [151, Theorem 2.9] provides us with precise conditions on radial convolution kernels K (more general than the Newtonian and the Bessel kernels) leading to a blowup of solutions of general diffusive aggregation equations with the Brownian diffusion of the form ut − Δu + ∇ ⋅ (u(∇K ∗ u)) = 0. They are strongly singular, that is, they have the singularity at 0: lim supx→0 x ⋅ ∇K(x) < 0, and are of moderate growth at ∞: |x ⋅ ∇K(x)| ≤ C|x|2 . Of course, the Bessel kernel Kγ is strongly singular in the sense of [151], as it is seen from (3.7.10). A discussion of related questions is in Chapter 5, Section 5.4, where aggregation equations are treated, see [42]. We prove Theorem 3.7.1 using the method of truncated moments, which is reminiscent of that in the papers [195], [163]. We will use a bump function ψ = (1 − |x|2 )2+ satisfying properties (3.3.3) and (3.6.6) and its rescalings for R > 0: ψR (x) = ψ( Rx ). The function ψ is piecewise 𝒞 2 (ℝ2 ), with supp ψ = {|x| ≤ 1}, and satisfies Δψ(x) = (−8 + 16|x|2 ) ≥ −8ψ(x) ≥ −8

for

|x| < 1.

(3.7.12)

We will use in the sequel the fact that ψ is strictly concave in a neighbourhood of x = 0, that is, Lemma 3.6.4. Next, we recall a well known property of concave functions. Lemma 3.7.8. For every function Ψ : ℝ2 → ℝ, which is strictly concave on a domain Ω ⊂ ℝ2 , we have for all x, y ∈ ℝ2 (x − y) ⋅ (∇Ψ(x) − ∇Ψ(y)) ≤ −θ|x − y|2 ,

(3.7.13)

where θ > 0 is the constant of strict concavity of Ψ on Ω, that is, satisfying HΨ ≤ −θ I with Hessian Hψ of the weight function ψ. Proof. By the concavity, we obtain Ψ(x) ≤ Ψ(y) + ∇Ψ(y) ⋅ (x − y) −

θ |x − y|2 . 2!

Summing this inequality with its symmetrized version (with x, y interchanged) leads to the claim.

3.7 Keller–Segel model with chemoattractant consumption terms | 121

Now, we formulate a crucial inequality in our proof of the blowup result. Lemma 3.7.9. For the Bessel kernel Kγ with γ ≥ 0 and a strictly concave function Ψ, we have for all x, y on the domain of the strict concavity of Ψ ∇Kγ (x − y) ⋅ (∇Ψ(x) − ∇Ψ(y)) ≥

θ g (|x − y|), 2π γ

(3.7.14)

where θ is the constant of the strict concavity of Ψ introduced in Lemma 3.7.8, and gγ is a radially symmetric continuous function, such that ∇Kγ (x) = −

1 x g (|x|). 2π |x|2 γ

(3.7.15)

In particular, gγ (0) = 1, the profile of gγ decreases, and gγ (|x|) ≤ Ce−√γ|x| . Proof. Combining Lemma 3.7.8 with equation (3.7.15) and properties (3.7.8), (3.7.9), and (3.7.10), we arrive immediately at the claimed formula. Now, we can prove our main blowup result with the approach already used in the proof of Theorem 3.6.1. Proof of Theorem 3.7.1. Similarly to the proof of Theorem 3.5.1, we consider the quantity wR (t) = ∫ u(x, t)ψR (x) dx, a local moment of u(., t), with the rescaled bump function ψR . By (3.6.14) MR (t) ≡ ∫{|x| 0. If M > 8π and u0 is sufficiently well concentrated near the origin, that is, θ(ε) g (2εR)wR (0) > k2 = 8 and, at the same time C(ε)(M − wR (0)) is sufficiently 4π γ small, then the solution u cannot be global-in-time. Case iii) In the case α < 2, the blowup occurs if for some R > 0 the quantity Rα−2 ∫{|x|0 ‖u(t)‖1 < ∞ as well as the optimal decay (hypercontractive) estimate supt>0 t 1−1/p ‖u(t)‖p < ∞. First of all, the mild solutions of the Cauchy problem (3.7.1)–(3.7.3) are studied via the integral equation (also known as the Duhamel formula) u(t) = etΔ u0 + ℬ(u, u)(t),

(3.7.22)

with the bilinear term ℬ, defined now as t

ℬ(u, z)(t) = − ∫(∇e

(t−s)Δ

) ⋅ (u(s) ∇(−Δ + γ)−1 z(s)) ds.

(3.7.23)

0

Then, to solve equation (3.7.22) in a Banach space (ℰ , ‖ ⋅ ‖ℰ ) of vector-valued functions, it is sufficient to prove the boundedness of the bilinear form ℬ : ℰ × ℰ → ℰ 󵄩 󵄩󵄩 󵄩󵄩ℬ(u, z)󵄩󵄩󵄩ℰ ≤ η‖u‖ℰ ‖z‖ℰ ,

(3.7.24)

with a constant η independent of u and z. The first and the second steps toward the proof of Theorem 3.7.5 are based on Lemma 1.1.1. 3.7.3 Step 1. Local-in-time solutions with the initial data in L1 Lemma 3.7.10. For every u0 ∈ L1 (ℝ2 ) and p ∈ (4/3, 2), there exists T > 0 independent of γ such that equation (3.7.22) has a solution u = u(x, t) in the space p

2

ℰ = {u ∈ Lloc ((0, T); L (ℝ )) : ∞

󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p < ∞},

0 0 so small to have 1 󵄩 󵄩 . sup t 1−1/p 󵄩󵄩󵄩etΔ u0 󵄩󵄩󵄩p < 4C 00

Proof. Let

1 r

=

2 p

+ 21 −

1 σ

(3.7.32)

for some suitable σ ∈ (1, p) so that r ∈ (1, 2). Moreover, denote

by q a number satisfying

1 p

+

1 q

= 1r . Under this choice of parameters, we make sure

3.7 Keller–Segel model with chemoattractant consumption terms | 127

that

1 p

−

1 q

< 21 , and q > 2, so that we can use (3.7.7) to estimate the bilinear form ℬ t

󵄩 󵄩󵄩 −1/2+1/p−1/r 󵄩 󵄩󵄩u(s)∇(−Δ + γ)−1 z(s)󵄩󵄩󵄩 ds 󵄩󵄩ℬ(u, z)(t)󵄩󵄩󵄩p ≤ C ∫(t − s) 󵄩r 󵄩 0

t

󵄩 󵄩 󵄩 󵄩 ≤ C ∫(t − s)2/σ−1/p−1 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p 󵄩󵄩󵄩∇(−Δ + γ)−1 z(s)󵄩󵄩󵄩q ds 0

≤ Cγ

−1/2−1/q+1/p

t

∫(t − s)1/σ−1/p−1 s2/p−2/σ 0

1/σ−1/p 󵄩 󵄩

󵄩󵄩 1/σ−1/p 󵄩 󵄩󵄩z(s)󵄩󵄩󵄩 ) ds 󵄩󵄩u(s)󵄩󵄩p )( sup s 󵄩p 󵄩

× ( sup s 0 0 independent of γ. Moreover, u( T2 ) ∈ Lσ (ℝ2 ) ⊂ L1 ∩Lp . Thus, we may continue this localin-time solution u(t) to the whole half-line (0, ∞) choosing γ > 0 sufficiently large. We notice that on the interval (T/2, T) solutions obtained in Lemma 3.7.10 and Lemma 3.7.12 coincide as a consequence of uniqueness assertion of Lemma 3.7.10. Proving optimal decay estimates remains. 3.7.5.1 The optimal L1 bound Next, we show that the solution satisfies the uniform global estimate 󵄩 󵄩 sup󵄩󵄩󵄩u(t)󵄩󵄩󵄩1 < ∞. t>0

(3.7.33)

For t ≥ T, similarly to the proof of Lemma 3.7.11, we consider a sequence of approximations of u, combine estimates (3.7.29) and an analog of (3.7.30) (this time on (T, ∞))

with 1 < σ < 4/3 < r < p < 2 < q, θ = ε=

1−

1−

1 σ 1 p

,

1 1 − r p

1− p1

. Moreover, we define

so that t 1/σ−1/p = t (1−1/p)(1−ε) .

(3.7.34)

128 | 3 Blowups Again, we arrive at an estimate similar to (3.7.27): t

󵄩2θ 󵄩2(1−θ) 󵄩󵄩 󵄩 󵄩󵄩 −1/2 󵄩 󵄩󵄩wn (s)󵄩󵄩󵄩1 ds, 󵄩󵄩wn+1 (t)󵄩󵄩󵄩1 ≤ ‖u0 ‖1 + ∫(t − s) 󵄩󵄩󵄩wn (s)󵄩󵄩󵄩p 0

where ϱ = 2θ < 1, since we can choose 2r < 1 + p1 . We split the integral on the right-hand side into two integrals over the intervals (0, T/2) and (T/2, t). The first one is estimated by C(T) in view of Lemma 3.7.10 and (3.7.27). To estimate the second interval, we notice that t

1 󵄩 󵄩2θ 󵄩2(1−θ) 󵄩󵄩 ∫ (t − s)− /2 󵄩󵄩󵄩wn (s)󵄩󵄩󵄩p 󵄩󵄩wn (s)󵄩󵄩󵄩1 ds

T/2

t

≤ CAn (t)ϱ ∫ (t − s)− /2 (s − 1

T/2

2(1−θ)(1/p−1/σ)

T ) 2

ds,

but t

1

∫ (t − s)− /2 (s − T/2

2(1−θ)(1/p−1/σ)

T ) 2

ds ≤ C(t −

1/2−2(1−1/r)(1−ε)

T ) 2

holds with ε as in (3.7.34), and the first inequality is obtained using (3.7.32). We notice that choosing σ > 1 close enough to 1, and r > 4/3 close enough to 4/3, we ensure that 1/2 − 2(1 − 1/r)(1 − ε) < 0. We proceed further as in the proof of Lemma 3.7.11 and arrive at (3.7.33). Remark 3.7.13. One can show by a standard method that u ∈ 𝒞 ([0, T); L1 (ℝ2 )). Here, it suffices to use the continuity of the bilinear form ℬ as in the proof in [37, Theorem 1.1]. 3.7.5.2 The optimal hypercontractive estimate for p > 1 First, we improve the decay estimates from Lemma 3.7.12. Lemma 3.7.14. For each p ∈ (4/3, 2), the solution of (3.7.1)–(3.7.2) with u0 ∈ L1 (ℝ2 ) satisfies 󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩u(⋅, t)󵄩󵄩󵄩p < ∞. t>0

(3.7.35)

Proof. By definition (3.7.31) of the space ℰ̃, one immediately sees that it is enough to prove (3.7.35) for t ≥ T. By the Duhamel formula (3.7.22), inequalities (40) and (3.7.7), we have t

󵄩 tΔ 󵄩 󵄩 󵄩󵄩 −1/2−1+1/p 󵄩 󵄩󵄩u(s)∇(−Δ + γ)−1 u(s)󵄩󵄩󵄩 ds 󵄩󵄩u(t)󵄩󵄩󵄩p ≤ 󵄩󵄩󵄩e u0 󵄩󵄩󵄩p + C ∫(t − s) 󵄩1 󵄩 0

3.7 Keller–Segel model with chemoattractant consumption terms | 129 t

1 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩etΔ u0 󵄩󵄩󵄩p + C ∫(t − s)− /2−1+1/p 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p 󵄩󵄩󵄩∇(−Δ + γ)−1 u(s)󵄩󵄩󵄩q ds

≤ Ct

1/p−1 T/2

0

(3.7.36)

‖u0 ‖1

t

3 3 󵄩2 󵄩2 󵄩 󵄩 + C ∫ (t − s)− /2+1/p 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p ds + C ∫ (t − s)− /2+1/p 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p ds.

0

T/2

Using (3.7.31), we estimate the second term on the right-hand side of (3.7.36) as T/2

3 󵄩 󵄩2 ∫ (t − s)− /2+1/p 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p ds

0

T/2

−3/2+1/p

T ) 2

≤ C ∫ (t − s)− /2+1/p s2(1/p−1) ds ≤ C(T)(t − 3

0

.

Hence for t ≥ T, and in view of the fact that for t ≥ T

it holds that

t T ≤t− , 2 2

(3.7.37)

relation (3.7.36) reads as follows: t

t

3 󵄩2 󵄩 󵄩 −1/2 1−1/p ∫ (t − s)− /2+1/p 󵄩󵄩󵄩u(s)󵄩󵄩󵄩p ds. 󵄩󵄩u(t)󵄩󵄩󵄩p ≤ C‖u0 ‖1 + Ct + Ct

1−1/p 󵄩 󵄩

T/2

In view of (3.7.31), and owing to definition of ε in (3.7.34), for t ≥ T, we arrive at t

T 3 󵄩 󵄩 t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p ≤ C‖u0 ‖1 + C + Ct 1−1/p ∫ (t − s)− /2+1/p (s − ) 2

−2(1−1/p)(1−ε)

ds.

(3.7.38)

T/2

Next, we use the inequality t

3

∫ (t − s)− /2+1/p (s − T/2

T ) 2

−2(1−1/p)(1−ε)

ds ≤ C(t −

1/p−1/2−2(1−1/p)(1−ε)

T ) 2

,

to see that by (3.7.37) for t ≥ T, (3.7.38) yields T 1 󵄩 󵄩 t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p ≤ C + Ct /2 (t − ) 2

−2(1−1/p)(1−ε)

1

≤ C + Ct /2−2(1−1/p)(1−ε) .

Since p ∈ (4/3, 2), it is enough to pick up ε > 0 small enough to ensure that 1 )(1 − ε) < 0, and we arrive at (3.7.35) for t ≥ T; Lemma 3.7.14 is proved. p

1 2

− 2(1 −

130 | 3 Blowups 3.7.5.3 The optimal decay estimate for other p ∈ (1, ∞) In view of Lemma 3.7.14, we see that Theorem 3.7.5 is true for p = 1 and p ∈ (4/3, 2). Since for p ∈ (1, 4/3], we have by interpolation ̄󵄩 󵄩 1−ϑ 󵄩ϑ 󵄩 󵄩 󵄩 t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p ≤ 󵄩󵄩󵄩u(t)󵄩󵄩󵄩1 (t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p̄ ) ,

where p̄ < 2, ϑ =

1 1 − p p̄ 1− p1̄

, and therefore Theorem 3.7.5 holds also for p ∈ (1, 4/3].

Now, we can interpolate estimate (3.7.35) for p ∈ (4/3, 2) and (3.7.33) to get (3.7.35) with any p ∈ [1, 2). The last step of the proof of Theorem 3.7.5 is the extrapolation of the hypercontractive estimates (3.7.35) for q ∈ [2, ∞). Actually, it is enough to obtain the decay estimate for q ∈ (2, ∞), the remaining case q = 2 will follow by simple interpolation. Taking q ∈ (2, ∞), we know that t

1 󵄩 󵄩 󵄩 󵄩󵄩 1/q−1 ‖u0 ‖1 + C ∫(t − s)− /2−1/r+1/q 󵄩󵄩󵄩u(s)∇(−Δ + γ)−1 u(s)󵄩󵄩󵄩r ds 󵄩󵄩u(t)󵄩󵄩󵄩q ≤ Ct

0

t

1 󵄩 󵄩 󵄩 󵄩 ≤ Ct 1/q−1 ‖u0 ‖1 + C ∫(t − s)− /2−1/r+1/q 󵄩󵄩󵄩u(s)󵄩󵄩󵄩σ 󵄩󵄩󵄩∇(−Δ + γ)−1 u(s)󵄩󵄩󵄩ρ ds

0

≤ Ct

1/q−1

t

1 󵄩2 󵄩 ‖u0 ‖1 + C ∫(t − s)− /2−1/r+1/q 󵄩󵄩󵄩u(s)󵄩󵄩󵄩σ ds.

(3.7.39)

0

Here, r is chosen in such a way that 1r = σ2 − 21 , so that for σ ∈ (4/3, 2) we have r ∈ (1, 2), r close to 2. At the same time, ρ1 + σ1 = 1r , and ρ1 = σ1 − 21 . The above choice of parameters

allows us to apply (3.7.7) to (3.7.39). Relation (3.7.35) with σ ∈ (4/3, 2) ‖u(s)‖σ ≤ Cs1/σ−1 , applied to (3.7.39), yields t

1 󵄩 󵄩󵄩 1/q−1 ‖u0 ‖1 + C ∫(t − s)− /2−1/r+1/q s2(1/σ−1) ds. 󵄩󵄩u(t)󵄩󵄩󵄩q ≤ Ct

0

Since t

∫(t − s)− /2−1/r+1/q s2(1/σ−1) ds = Ct − /2−1/r+1/q+2/σ−1 , 1

1

0

we notice that ‖u(t)‖q ≤ Ct 1/q−1 ‖u0 ‖1 + Ct 1/q−1 holds in view of the relation 1 1 1 2 1 − − + + − 1 = − 1. 2 r q σ q Thus, the decay estimate for q > 2 is proved.

3.8 Blowup of radially symmetric solutions, again

| 131

3.8 Blowup of radially symmetric solutions, again Here, we formulate new sufficient conditions (based on the analysis of the dα -concentration of solutions) for the nonexistence of global-in-time radial solutions to problem with the fractional diffusion α

ut + (−Δ) /2 u + ∇ ⋅ (u∇v) = 0, Δv + u = 0,

x ∈ ℝd , t > 0,

(3.8.1)

d

x ∈ ℝ , t > 0,

(3.8.2)

in ℝd , supplemented with the nonnegative initial condition u(x, 0) = u0 (x).

(3.8.3)

Theorem 3.8.1 (Blowup of solutions). Let α ∈ (0, 2]. Consider a local-in-time, nonnegative, classical, radially symmetric solution u ∈ 𝒞 ([0, T), L1loc (ℝd )) of problem (3.8.1)– (3.8.3) with a nonnegative radially symmetric initial datum u0 ∈ L1loc (ℝd ). There exists a constant cα,d > 0 such that i) if sup Rα−d ∫ u0 (x) dx > cα,d , R>0

(3.8.4)

{|x| 0. ii) If, moreover, lim sup Rα−d ∫ u0 (x) dx > cα,d , R→0

(3.8.5)

{|x| 0. Remark 3.8.2. Condition (3.8.4) means that the Morrey space norm in M /α (ℝd ) of the initial datum u0 is large enough, see Proposition 2.7.1. Moreover, condition (3.8.5) applies only to initial conditions, which are singular at the origin. This condition implies that problem (3.8.1)–(3.8.3) is ill posed in the vicinity of such u0 in the space d 𝒞 ([0, T], M /α (ℝd )) for every T > 0. We return to this topic in Chapter 5, Section 5.2. d

Remark 3.8.3. Notice that Theorem 3.8.1 also holds true for α = 2. In this case, results in Theorem 3.8.1 are generalizations and improvements, whereas their proofs are simplifications of those in previous Sections 3.2, 3.5, and [50, 51], where problem (3.8.1)– (3.8.3) with α = 2 was considered. In particular, the estimate for the number c2,d proved in [51, Theorem 1.1] was twice worse than that one in this work, see Remark 3.7.3 for more detail.

132 | 3 Blowups We prove Theorem 3.8.1 using again the modified method of truncated moments, which is reminiscent of that in the papers [195, 163, 50] for α = 2 and in our recent papers [50, 51, 28], adjusted to the case α < 2. First, we define a new continuous bump function ψ, which is better suited to problems with fractional Laplacians, and its rescalings for R > 0 1+α/2

ψ(x) = (1 − |x|2 )+

(1 − |x|2 )1+ /2 ={ 0 α

for for

|x| < 1, |x| ≥ 1,

x ψR (x) = ψ( ). R

(3.8.6)

The function ψ is piecewise 𝒞 2 (ℝd ), with its support supp ψ = {|x| ≤ 1}, and satisfies α/2

∇ψ(x) = −(α + 2)x(1 − |x|2 )+ .

(3.8.7)

α

The action of the fractional powers (−Δ) /2 of the Laplacian operator on functions, such as (1 − |x|2 )κ+ leads to explicit formulas, involving hypergeometric functions. In the particular case κ = 1 + α/2, it follows from [183, p. 39] that this is a linear polynomial in |x|2 α

(−Δ) /2 ψ(x) = mα (1 −

d+α 2 |x| ) on d

{|x| ≤ 1},

(3.8.8)

with the constant mα = 2α Γ(2 +

d+α

α Γ( 2 ) α α α α ≈ 2 /2 (1 + )Γ(1 + )d /2 ) 2 Γ( d ) 2 2 2

as d → ∞.

(3.8.9)

The relation for quotients of the Gamma functions used to obtain asymptotics of mα Γ(z + a) ≍ z a−b Γ(z + b)

as z → ∞,

(3.8.10)

follows directly from the Stirling formula (see [236]) Γ(z + 1) ≈ √2πz z z e−z

as

z → ∞.

(3.8.11)

Moreover, α

(−Δ) /2 ψ(x) ≤ 0,

|x| ≥ 1,

holds similarly as was shown in [28, Lemma 4.3], so that we have the inequality α

(−Δ) /2 ψ(x) ≤ ℓα ψ(x)

(3.8.12)

with 2 /2 Γ(1 + α2 ) (d + α)1+α/2 Γ( d+α Γ(1 + α2 ) α ) 2 ≈ d d (1 + α2 )α/2 (1 + α2 )α/2 Γ( d2 ) α

ℓα =

as d → ∞.

(3.8.13)

3.8 Blowup of radially symmetric solutions, again

| 133

Indeed, ℓα is the least number such that the inequality ℓα (1 − s)1+ /2 − mα (1 − α

d+α s) ≥ 0 d

holds for each s ∈ [0, 1], the minimum of that expression being attained at s0 = [0, 1). Now, consider a “local moment” of the solution u(., t) defined by wR (t) = ∫ ψR (x)u(x, t) dx

d−2 d+α

∈

(3.8.14)

with the weight function ψ as in (3.8.6). The evolution of wR is determined by d α w (t) = − ∫ ψR (x)(−Δ) /2 u(x, t) dx + ∫ u(x, t)∇v(x, t) ⋅ ∇ψR (x) dx dt R α

= − ∫(−Δ) /2 ψR (x) u(x, t) dx 󵄨󵄨 x 󵄨󵄨2 α/2 󵄨 󵄨 − (α + 2) ∫ u(x, t)∇v(x, t) ⋅ xR−2 (1 − 󵄨󵄨󵄨 󵄨󵄨󵄨 ) dx 󵄨󵄨 R 󵄨󵄨 + 2 d + α |x| ≥ −mα R−α ∫ (1 − )u(x, t) dx d R2

(3.8.15)

{|x|≤R}

α/2

|x|2 ∫ u(x, t)(∇v(x, t) ⋅ x)(1 − 2 ) dx. R

− (α + 2)R

−2

{|x|≤R}

The second equality followed from the “integration by parts” for the selfadjoint operα ator (−Δ) /2 . Thus, applying inequality (3.8.12) and Lemma 2.1.8, we obtain d w (t) ≥ −ℓα R−α wR (t) dt R +

(3.8.16) α/2

α + 2 −d |x|2 R ∫ (1 − 2 ) u(x, t)M(|x|, t)|x|2−d dx. σd R {|x|≤R}

Let us write the terms on the right-hand side of inequality (3.8.15) in the radial variables explicitly. We have 1

1+α/2

wR (t) = R ∫ M 󸀠 (Rr, t)(1 − r 2 )

1

α/2

dr = (α + 2) ∫ M(Rr, t)r(1 − r 2 ) dr,

0

0

and likewise, after the integration by parts, 1

α/2

R ∫ M 󸀠 (Rr, t)M(Rr, t)r 2−d (1 − r 2 ) dr 0

1

α/2−1 1 ((d − 2) − (d − 2 − α)r 2 ) dr. = ∫ M(Rr, t)2 r 1−d (1 − r 2 ) 2

0

(3.8.17)

134 | 3 Blowups Now, the application of the Cauchy–Schwarz inequality shows that 1

α/2 1 wR (t) ≤ 2(α + 2) ∫ r 1+d (1 − r 2 ) dr d−2

2

0

1

×

α/2−1 α+2 ((d − 2) − (d − 2 − α)r 2 ) dr. ∫ M 2 r 1−d (1 − r 2 ) 2

0

Therefore, the inequality d w (t) ≥ − ℓα R−α wR (t) dt R +

1

α/2−1 (α + 2)R−d ((d − 2) − (d − 2 − α)r 2 ) dr ∫ M(Rr, t)2 r 1−d (1 − r 2 ) 2σd

0

implies d R−d wR (t) ≥ wR (t)(−ℓα R−α + C(α, d)wR (t)) dt σd

(3.8.18)

for some constant C(α, d) > 0. For the computation of C(α, d), we used above the relations (1 − r 2 )((d − 2) − (d − 2 − α)r 2 ) and

1

∫r 0

1+d

−1

≤ (d − 2)−1 ,

1

α d 1 1 Γ( 2 + 1)Γ(1 + 2 ) α/2 d/2 (1 − r ) dr = ∫ τ (1 − τ) dτ = , 2 2 Γ( d+α + 2) 2 α/2

2

0

the latter following from the definition of the Euler Beta function (1.4.4). Now, if initially Rα−d wR (0) > σd

ℓα , C(α, d)

(3.8.19)

then wR (t) strictly increases in time, and wR (t) blows up in a finite time, which is a contradiction if u(x, t) is a global-in-time radially symmetric, nonnegative solution. Now, the proof of Theorem 3.8.1(i) is complete, because of an obvious inequality ∫{|x| 0. Next, under condition (3.8.19), inequality (3.8.18) implies that d w w−1 ≥ ηR−α dt R R for some η > 0. Consequently, wR (t) ≥ wR (0) exp(ηR−α t).

(3.8.20)

3.8 Blowup of radially symmetric solutions, again

| 135

Under assumption (3.8.5), there exist a constant C > 0 and a sequence Rn → 0 such α −α that wRn (t) ≥ CRd−α n exp(ηRn t). Thus, we obtain wRn (t) > M for t ≥ T ≍ Rn asymptotically when Rn → 0, which implies that locally bounded u(r, t) cannot be defined on any interval [0, T] with some T > 0. Remark 3.8.4. A sufficient condition (3.8.19) for blowup can be expressed for α ≥ 1 in d terms of the Morrey norm of M /α (ℝd ), and we estimate that critical quantity sufficient for blowup asymptotically as d → ∞. Observe that σd

ℓα α ≈ Cα σd d /2−2 C(α, d)

asymptotically as

d→∞

(by (3.8.13), (3.8.18)) with Cα = 2

2+α/2

1−α/2

α (1 + ) 2

Γ(1 +

2

α ) ≤ 8. 2

(3.8.21)

The dα -radial concentration of u0 ≥ 0 (19) appearing in the assumptions of Theorem 3.8.1, that is, ||u0 || d ≡ sup Rα−d ∫ u0 (y) dy, R>0

α

{|y|0 Rα−d ∫ ψR (y)u0 (y) dy are equivalent. Indeed, for every locally integrable function ω ≥ 0 and all R > 0 and s ∈ (0, 1), we have ∫ ψR (x)ω(x) dx ≥

(1 − (1 − s))

∫

1+α/2

ω(x) dx

{|x|≤R√1−s}

= s1+ /2 α

ω(x) dx

∫ {|x|≤R√1−s}

and 1+α/2

max s

s∈[0,1]

(1 − s)

(d−α)/2

1+α/2

α+2 =( ) d+2

d−α ) d+2

(d−α)/2

(

≡ Hd .

Therefore, supR>0 Rα−d ∫ ψR ω ≥ Hd L if ||ω|| d > L, that is, the upper bound of the moα

ments, the radial concentration ||ω|| d of ω as well as the M /α (ℝd ) Morrey space norm d

α

|ω||M d/α for α ≥ 1 are comparable by Proposition 2.7.1. Note that we have asymptotically Hd−1 ≈ (

1+α/2

de ) α+2

.

Thus, condition (3.8.19) is satisfied if, for example, α ̃ dα−1 σ ||u0 || d > Cα d /2−2 σd Hd−1 ≈ C α d α

(3.8.22)

136 | 3 Blowups with Cα as in (3.8.21), and where 2

̃ = 2(1 + α ) Γ(1 + α ) e1+α/2 ≤ 2(1 + α ) e1+α/2 C α 2 2 2 −α

−α

—and this leads to a blowup. Therefore, we established that the asymptotic discrepancy between the critical quantity of the radial concentration for the global-in-time existence of solutions in Theorem 2.3.1, and the bound on the radial concentration guaranteeing σd s(α,d) d−α α the finite time blowup, is of order d /2 , because of relations (3.8.22) and (1.3.6), that is, α Γ(α) α/2−1 σd ||uC || d = d−α s(α, d) ≈ 2 /2 Γ( . α σd d ) α

2

4 Blowups à la Fujita 4.1 Fujita’s idea of the proof of blowup for the system with the classical diffusion This is yet another approach to prove blowup, based on the seminal idea in [125] for nonlinear heat equation with power nonlinearities. This permits us to obtain finer sufficient conditions for blowup of solutions in terms of Morrey space norms, even in a more direct way as before. We consider radially symmetric solutions of the classical Keller–Segel system ut − Δu + ∇ ⋅ (u∇v) = 0,

Δv + u = 0,

x ∈ ℝd , t > 0, d

x ∈ ℝ , t > 0,

(4.1.1) (4.1.2)

supplemented with the initial condition u(x, 0) = u0 (x) ≥ 0.

(4.1.3)

Even if in applications u0 ∈ L1 (ℝd ), and then mass M = ∫ℝd u0 (x) dx = ∫ℝd u(x, t) dx is conserved, we will be able to consider locally integrable solutions with infinite mass, such as the famous Chandrasekhar steady state singular solution (4), and “big”— polynomially bounded—solutions. Our main result in this section is a sufficient condition on radial initial data, which leads to a finite time blowup of solutions expressed in terms of quantities related d to the Morrey space norm M /2 (ℝd ) in Theorem 4.1.1. For instance, condition (4.1.18): TΔ supT>0 Te u0 (0) > C(d) for some C(d) ∈ [1, 2] is sufficient for the blowup of solution with the initial condition u0 . Sufficient blowup conditions expressed in terms of the radial concentration (20): ||u0 || > 𝒩 , together with an asymptotics of the number 𝒩 as d → ∞, are also in Proposition 4.1.6. Similar results for the system with modified diffusion operator α

ut + (−Δ) /2 u + ∇ ⋅ (u∇v) = 0, Δv + u = 0,

x ∈ ℝd , t > 0, d

x ∈ ℝ , t > 0,

(4.1.4) (4.1.5)

supplemented with the initial condition u(x, 0) = u0 (x) ≥ 0

(4.1.6)

will be derived and discussed in Section 4.2. In a parallel way, we also have – blowup of radial solutions with large initial data (Theorem 4.2.1, α ∈ (0, 2)); – a reformulation of sufficient condition for blowup of radial solutions in terms of d Morrey space M /α (ℝd ) norm (Proposition 4.2.2). https://doi.org/10.1515/9783110599534-004

138 | 4 Blowups à la Fujita For the proof of the main result, we revisit a classical argument of H. Fujita (applied to the nonlinear heat equation in [125] and extended in [229] to the case of fractional Laplacians in the diffusion terms) and reminiscent of ideas in [80], which leads to a sufficient condition for blowup of radially symmetric solutions of system (4.1.1)– (4.1.2), with a significant improvement compared to [52], where local moments of solutions have been employed. Then, we derive as corollaries of condition (4.1.18) other criteria for blowup of solutions of (4.1.1)–(4.1.3). Theorem 4.1.1. Let d ≥ 2. If the inequality TeTΔ u0 (0) > C(d) holds with an explicit constant C(d) ∈ [1, 2], see (4.1.15), then every radial (either classical or weak) solution of problem (4.1.1)–(4.1.3), which exists on [0, T] blows up in L∞ not later than t = T, that is, limt↗T ‖u(t)‖∞ = ∞. Proof. We had one parameter family of moments in the proofs of results in the previous Chapter 3, Sections 3.3, 3.5 with R > 0 as a parameter. Here, the anticipated time of blowup T is a parameter that might be adjusted. So, for a fixed T > 0, consider the weight function G = G(x, t), x ∈ ℝd , t ∈ [0, T), which solves the backward heat equation with the unit measure as the final time condition Gt + ΔG = 0,

(4.1.7)

G(⋅, T) = δ0 .

(4.1.8)

Clearly, we have a (unique nonnegative) solution G(x, t) = (4π(T − t))

−d/2

exp(−

|x|2 ), 4(T − t)

(4.1.9)

defined by the Gauss–Weierstrass kernel, satisfying ∫ G(x, t) dx = 1, so that ∇G(x, t) = −

x G(x, t). 2(T − t)

(4.1.10)

Define for a solution u of (4.1.1)–(4.1.2), which is supposed to exist on [0, T] the moment W(t) = ∫ G(x, t)u(x, t) dx.

(4.1.11)

Since G decays exponentially fast in x as |x| → ∞, the moment W is well defined (at least) for solutions u = u(x, t), which are polynomially bounded in x. The evolution of the moment W is governed by the identity dW = ∫ Gut dx + ∫ Gt u dx dt = ∫(Δu − ∇ ⋅ (u∇v))G dx − ∫ ΔG u dx = ∫ ΔG u dx + ∫ u∇v ⋅ ∇G dx − ∫ ΔG u dx

4.1 Fujita’s idea of the proof of blowup for the system with the classical diffusion

| 139

1 ∫ u∇v ⋅ xG dx 2(T − t) 1 = ∫ u(x, t)M(|x|, t)|x|2−d G(x, t) dx 2σd (T − t)

=−

(4.1.12)

∞

σd 1 = ∫ Mr (r, t)r 1−d M(r, t)r 2−d G(r, t)r d−1 dr 2σd (T − t) σd 0 ∞

=

1 ∫ Mr Mr 2−d G dr 2σd (T − t) 0

∞

=−

1 ∫ M 2 (r 2−d G)r dr, 4σd (T − t) 0 ∞

=

r2 1 )G dr, ∫ M 2 r 1−d ((d − 2) + 4σd (T − t) 2(T − t) 0

where we used the radial symmetry of the solution u in (4.1.12), Lemma 2.1.8 and, of course, the radial symmetry of the weight G. Expressing W in the radial variables, we obtain ∞

W(t) = σd ∫ 0 ∞

1 M r 1−d Gr d−1 dr σd r

= − ∫ MGr dr 0 ∞

= ∫M 0

r G dr. 2(T − t)

(4.1.13)

Now, applying the Cauchy–Schwarz inequality to the quantity (4.1.13), we get ∞

W 2 (t) = ( ∫ M 0 ∞

2

r G dr) 2(T − t)

≤ ∫ M 2 r 1−d ((d − 2) + 0

r2 )G dr 2(T − t)

(4.1.14)

1 r d+1 G × dr. ∫ 2 2(T − t) r + 2(d − 2)(T − t) ∞

0

Returning to the time derivative of W in identities (4.1.12), we arrive at the differential inequality

140 | 4 Blowups à la Fujita dW r d+1 1 G ≥ W 2 (t)( ∫ dr) dt 4σd (T − t) 2(T − t) r 2 + 2(d − 2)(T − t) ∞

−1

0

−1

∞

d/2

2 π −1 = W 2 (t)( ∫ ϱd+1 (2(d − 2) + 4ϱ2 ) e−ϱ dϱ) , 8σd

0

where ϱ =

r 1 . 2(T−t) /2

Recalling formula (11) for the area of the unit sphere, we denote C(d) =

16

∞

2

∫ ϱd+1 (2(d − 2) + 4ϱ2 ) e−ϱ dϱ. −1

Γ( d2 ) 0

(4.1.15)

Clearly, C(2) = 2, and C(d) < 2 for d ≥ 3, since we have C(d)


then lim supt↗T W(t) = ∞, which means: lim supt↗T, x∈ℝd u(x, t) = ∞—a contradiction with the existence of locally bounded solution u on the whole time interval [0, T]. Indeed, since limt↗T sup|x|>R G(x, t) = 0 for every R > 0, we get for every R > 0 lim sup sup u(x, t) = ∞, t↗T

|x| 0. Indeed, 1 t 1 − ≥ . W(0) W(t) C(d) So, if W(0) ≥

C(d) , T

then W(t) ≥

1 W(0)

1 −

t C(d)

≥

C(d) . T −t

4.1 Fujita’s idea of the proof of blowup for the system with the classical diffusion

| 141

For other results on blowup rates (also on a faster blowup, the so called, type II blowup), see [132, 187, 188, 189], which are, in a sense, parallel to those for nonlinear heat equation ut = Δu + u2 (as well as a more general equation ut = Δu + up ) studied in [213], and also Remark 5.7.2. Remark 4.1.3. A better bound for the constant C(d) (which might be computed using the incomplete Gamma function) than C(d) ∈ (1, 2) shown above is 2

1/2 Γ ( d+1 Γ ( d+1 ) ) d−1 2 2 2 2 ) < C(d) < ( < ( ) . d−1 d−2 d−2 Γ ( d2 ) Γ ( d2 )

(4.1.19)

To prove the upper bounds, observe that by the inequalities between harmonic, geometric and arithmetic means, the denominator of the integrand in (4.1.15) satisfies 1 1 1 1 1 + 2) ≤ ≤ ( 2 2(d − 2) + 4ϱ 4√2(d − 2)ϱ 4 2(d − 2) 4ϱ with a strict inequality whenever 2(d − 2) ≠ 4ϱ2 . Then, we have C(d)
2 𝜕r T>0 4π

0

for the radial distribution function M of the initial condition u0 . This means: supr>0 M(r) > 8π, and the well known blowup condition for radially symmetric solutions in ℝ2 is recovered in that way.

142 | 4 Blowups à la Fujita Observe that the equality in the Cauchy–Schwarz inequality (4.1.14) holds if and only if 0 ≤ M(r, t) =

r2

A(t)r d A(t)2d ϱd d = (T − t) /2−1 2 + 2(d − 2)(T − t) 4ϱ + 2(d − 2)

with some A(t) ≥ 0. Then inequality (4.1.17) reads W(t) = (

1 t − ) , W(0) C(d) −1

(4.1.20)

and if d ≥ 3 d 2 A(0) Γ( 2 ) A(0)r d+1 1 C(d) d e−r /(4T) (4πT)− /2 dr = , W(0) = C(d) ≥ ∫ 2 d/2 2T r + 2(d − 2)T T 8π T ∞

0

then the solution blows up not later than T. This holds exactly when A(0) ≥ 4σd since (11). This solution (see [80, (33)] and also [98]) satisfies identity (4.1.20) with W(0) = C(d) , and it is, in a sense, a kind of the minimal smooth blowing up solution. So, we T have Corollary 4.1.5. Moreover, if A(t) ≡ 4σd , d ≥ 3, we have an explicit example of blowing up solution with infinite mass M(r, t) =

r2

4σd r d , + 2(d − 2)(T − t)

(4.1.21)

= 2uC (x), that is, twice the singular stationary solution whose density approaches 4(d−2) |x|2 when t ↗ T, so that the density of this solution becomes infinite at the origin for t = T. Clearly, for this solution u and the corresponding initial density u0 , we have for each t ∈ [0, T) u0 (x) = 4(d − 2)

(r 2

r 2 + 2T , + 2(d − 2)T)2

and therefore |u0|M d/2 = 4σd = lim r 2−d M(r, t) = |u(t)||M d/2 . r→∞

We express below a sufficient condition (4.1.18) for blowup in terms of the radial concentration. Proposition 4.1.6 (Comparison of blowing up solutions). Let d ≥ 3, and define the threshold number: 𝒩 = inf{N : solution with the initial datum satisfying M(r) = N 1[1,∞) (r)

blows up in a finite time}.

4.1 Fujita’s idea of the proof of blowup for the system with the classical diffusion | 143

Then the asymptotic relation 𝒩 ⪅ 4σd √π(d − 2)

holds as d → ∞. Therefore, if u0 ≥ 0 is such that ||u0 || > 𝒩 , then the solution with u0 as initial datum blows up in a finite time. The inequality ||u0 || > 𝒩 means that the radial distribution function corresponding to such u0 satisfies M(r) ≥ Rd−2 𝒩 1[R,∞) (r) for some R > 0. Above, the radial distribution function, 1[1,∞) corresponds, of course, to the normalized Lebesgue measure σd−1 dS on the unit sphere 𝕊d−1 . Proof. Here, and in the sequel, due to the scaling properties of system (4.1.1)–(4.1.2), we may consider R = 1, which does not lead to loss of generality. First note that if u0 (x) ≥ 0 is such that N = M(1, 0) > 𝒩 for the corresponding radial distribution function M, then M(r, 0) ≥ N 1[1,∞) (r) for all r > 0, and the solution u with u0 as the initial datum blows up in a finite time. Indeed, this is an immediate consequence of the averaged comparison principle, that is, [46, Theorem 2.1] or the direct comparison principle for equation (4.1.22) below (which is, however, a nontrivial property), see also [20] for an analysis in the case d = 2 𝜕M d−1 1 = Mrr − Mr + r 1−d MMr . 𝜕t r σd

(4.1.22)

The above equation is, of course, equivalent to (10). Thus, from equation (4.1.21), we know that if sup tetΔ u0 (0) > 2 = 2 sup tetΔ ( t>0

t>0

2(d − 2) )(0), |x|2

(4.1.23)

then u blows up in a finite time. To check that K2 (d) ≡ sup tetΔ ( t>0

2(d − 2) )(0) = 1, |x|2

(4.1.24)

let us compute d

t(4πt)− /2 ∫ |x|−2 exp(−

2 |x|2 1 d ) dx = π − /2 ∫ |z|−2 e−|z| dz 4t 4

∞

=

2 1 −d/2 π σd ∫ e−ϱ ϱd−3 dϱ 4

0

=

1

∞

d

∫ e−τ τ /2−2 dτ

4Γ( d2 ) 0

144 | 4 Blowups à la Fujita 1 d Γ( − 1) 4Γ( d2 ) 2 1 . = 2(d − 2)

=

(4.1.25)

Note that, by the above computations, there exist radial initial data u0 ∈ L1 (ℝd ) ∩ L∞ (ℝd ) with W(0) as close to 1, as we wish. In other words, we have C(d) ∈ [1, 2). To calculate the asymptotics of the number 𝒩 , observe that the quantity supt>0 tetΔ u0 (0) in (4.1.23) for the normalized Lebesgue measure σd−1 dS on the unit sphere 𝕊d−1 is equal to −d/2 − 4t1

L2 (d) ≡ sup t(4πt) t>0

e

d/2−1

1 d d−2 ) = π − /2 ( 4 2

e1− /2 . d

(4.1.26)

Therefore, by (11) and the Stirling formula (3.8.11) for the Gamma function, the asympand 𝒩 ⪅ 4σd √π(d − 2) hold. totic relations L2 (d) ≈ 2σ1 √ 1 d

π(d−2)

This substantially improves the estimate of 𝒩 ≍ dσd in Chapter 3, Section 3.5 (see Remark 3.5.2) first derived in [52, Section 8]. We give (below) some other examples of initial data leading to a finite time blowup of solutions. Remark 4.1.7 (Examples of blowing up solutions). Observe that, evidently, for each initial condition u0 ≢ 0, there is N > 0 such that condition (4.1.18) is satisfied for Nu0 . Clearly, by ||uC || = |uC|M d/2 = 2σd and identities (4.1.23)–(4.1.25), for each η > 2, the solution with the initial condition u0 = ηuC blows up. Moreover, for each η > 2 and sufficiently large R = R(η) > 1, the bounded initial condition of compact support u0 = η1{1≤|x|≤R} uC leads to a blowing up solution, see (4.1.18). The singularity of that solution at the blowing-up time is ≍ |x|1 2 at the origin. It seems that the latter result cannot be obtained applying previously known sufficient criteria for blowup, such as (3.1.1). On the other hand, the initial data, such as min{1, uC } + ϵψ with any smooth nonnegative, compactly supported function ψ and a sufficiently small ϵ > 0 (they are somewhere above the critical uC pointwisely) still lead to global-in-time solutions, according to [46, Theorem 2.1]. Remark 4.1.8 (Equivalent qualitative conditions for blowup). The condition ℓ(u0 ) = supt>0 tetΔ u0 (0) > 2 is sufficient for blowup, see condition (4.1.23). The quantity ℓ̃ ≥ ℓ ̃ 0 ) ≡ sup t 󵄩󵄩󵄩etΔ u0 󵄩󵄩󵄩 ℓ(u 󵄩∞ 󵄩 t>0

(4.1.27)

d is a Banach space norm equivalent to the norm of the Besov space B−2 ∞,∞ (ℝ ). Thus, tΔ for nonnegative functions u0 the property supt>0 t‖e u0 ‖∞ ≫ 1 is equivalent to the condition |u0|M d/2 ≫ 1, see [172, Proposition 2 B)]. Note that, however, the comparison

4.2 Blowup à la Fujita of solutions of system with fractional diffusion | 145

constants for ℓ, ℓ̃ and || ⋅ ||, | ⋅ |M d/2 strongly depend on d, see [52, Proposition 7.1, Remark 8.1]. In summary, qualitative sufficient conditions for blowup for radial u0 ≥ 0 – supt>0 tetΔ u0 (0) ≫ 1, – supt>0 t‖etΔ u0 ‖∞ ≫ 1, – ||u0 || ≡ supr>0 r 2−d ∫{|x|0, x∈ℝd r 2−d ∫{|y−x| 2 (so that by Proposition 4.1.6 condition 2√πd 2σd ⪅ ||u0 || asymptotically guarantees that), then the solution of problem (4.1.1)–(4.1.3) blows up not later than at t = T. Remark 4.1.10 (A general problem). In view of the above remarks, the following question is left open: What may happen with the solution u of (4.1.1)–(4.1.2) with u0 in (4.1.3) satisfying 2σd < ||u0 || ≤ 4σd ?

4.2 Blowup à la Fujita of solutions of system with fractional diffusion Here, we generalize results for the Brownian diffusion case α = 2 to the case of the system of nonlocal diffusion-transport equations (4.1.4)–(4.1.6), generalizing the classical Keller–Segel system of chemotaxis to the case of the diffusion process given by α the fractional power of the Laplacian (−Δ) /2 with α ∈ (0, 2), a nonlocal operator, as was in [28, 52]. d The generalized Chandrasekhar discontinuous solution uC ∈ M /α (ℝd ) is, in a sense, a critical one, which is not smoothed out by the diffusion operator in system (4.1.4)–(4.1.5). As usual for nonlinear evolution equations of parabolic type, blowup of a solution u at t = T means: lim supt↗T, x∈ℝd u(x, t) = ∞. In fact, some Lp norms (with p ≥ d/α) of d u(t) blow up together with the L∞ -norm, but not the critical M /α (ℝd ) norm. α/2

Theorem 4.2.1 (Blowup of solutions). If d ≥ 2, α ∈ (0, 2), Te−T(−Δ) u0 (0) > Cα (d) for a constant Cα (d) > 0, defined below in (4.2.7), then each solution of problem (4.1.4)–(4.1.6) blows up not later than t = T.

146 | 4 Blowups à la Fujita The proof of Theorem 4.2.1 does not apply to the case d = 1, which is studied by completely different methods in [76], see also [82, 83, 84]. Informally speaking, this sufficient condition for blowup is equivalent to |u0|M d/α ≫ 1. Indeed, for radially symmetric nonnegative functions, the condition α/2

supT>0 Te−T(−Δ) u0 (0) ≫ 1 is equivalent to the relation α/2 󵄩 󵄩 sup󵄩󵄩󵄩Te−T(−Δ) u0 󵄩󵄩󵄩∞ ≫ 1.

T>0

α/2

This fact can be proved using fine estimates of the kernel of the semigroup e−t(−Δ) restricted to radial functions, as was in the case α = 2, see Remark 5.1.2. Further, the α −T(−Δ) /2 quantity supT>0 ‖Te u0 ‖∞ is equivalent to the Morrey space norm |u0|M d/α . Moreover, we note a useful characterization of the homogeneous Besov spaces α/2 󵄩 󵄩 sup T 󵄩󵄩󵄩e−T(−Δ) u󵄩󵄩󵄩∞ < ∞

d if and only if u ∈ B−α ∞,∞ (ℝ ),

T>0

shown in [172, Proposition 2B)] for α = 2, and for α ∈ (0, 2) derived from [174, Section 4, proof of Proposition 2]. Proof. As in the original reasoning of Fujita in [125] applicable to the case α = 2 (in Section 2), and in [229] for a nonlinear fractional heat equation with power source terms and α ∈ (0, 2), here we consider the moment W(t) = ∫ G(x, t)u(x, t) dx

(4.2.1)

with the weight function G = G(x, t) solving the backward linear fractional heat equation on (0, T) α

Gt − (−Δ) /2 G = 0,

(4.2.2)

G(⋅, T) = δ0 .

(4.2.3)

It is clear that G has the selfsimilar radially symmetric form d

G(x, t) = PT−t,α (x) = (T − t)− /α R(

|x| ) (T − t)1/α

(4.2.4)

with the same function R as in (34), satisfying properties (35), (36), (37). For radially symmetric functions, W in (4.2.1) becomes −d/α

W(t) = −(T − t)

∞

1

∫ M(r, t)R󸀠 (ϱ)(T − t)− /α dr 0

with

r

M(r, t) = ∫ u(x, t) dx = σd ∫ u(ϱ, t)ϱd−1 dϱ {|x| 0, so that the integral ∫{1≤|y|}

u(x−y,t) |y|d+α

dy must be finite.

Furthermore, we have by Lemma 2.1.8 that ∇v(x)⋅x = − σ1 |x|2−d ∫{|y|≤|x|} u(y) dy, and d

α

therefore by selfadjointness of the operator (−Δ) /2 dW = ∫ Gut dx + ∫ Gt u dx dt

α

α

= ∫(−(−Δ) /2 u − ∇ ⋅ (u∇v))G dx + ∫(−Δ) /2 Gu dx = ∫ u∇v ⋅ ∇G dx

(4.2.5) ∞

=−

d+1 𝜕 1 (T − t)− α ∫ MMr 1−d R󸀠 (ϱ) dr σd 𝜕r

0 ∞

=

d+1 1 M 2 𝜕 1−d 󸀠 (r R (ϱ)) dr. (T − t)− α ∫ σd 2 𝜕r

0

Using the Cauchy–Schwarz inequality as in preceding Section 4.1, we estimate 2

− d+1 α

W (t) ≤ (T − t)

∞

∫ 0

− d+1 α

× (T − t)

󵄨󵄨 M 2 󵄨󵄨󵄨󵄨 𝜕 1−d 󸀠 󵄨 󵄨󵄨 (r R (ϱ))󵄨󵄨󵄨 dr 󵄨󵄨 2σd 󵄨󵄨 𝜕r ∞

∫ 2σd 0

|R󸀠 (ϱ)|2

| 𝜕r𝜕 (r 1−d R󸀠 (ϱ))|

dr.

(4.2.6)

Note that the function ϱ1−d R󸀠 (ϱ) is strictly decreasing as the product of two strictly decreasing positive functions, so that the denominator of the integrand in (4.2.6) is strictly positive. Now, with the definition of Cα (d) ∞

Cα (d) = 2σd ∫

|R󸀠 (ϱ)|2

| 𝜕 (ϱ1−d R󸀠 (ϱ))| 0 𝜕ϱ

dϱ,

the differential inequality obtained from relations (4.2.5) and (4.2.6) dW 1 ≥ W 2 (t) dt Cα (d)

(4.2.7)

148 | 4 Blowups à la Fujita leads to the estimate 1 1 T − ≥ . W(0) W(T) Cα (d) Thus, a sufficient condition for the blowup becomes α/2

Te−T(−Δ) u0 (0) > Cα (d).

(4.2.8)

Indeed, if (4.2.8) holds, then W(t) ≥

1 W(0)

1

≥

t Cα (d)

−

Cα (d) , T −t

and limt↗T W(t) = ∞, and therefore lim supt↗T u(x, t) = ∞. Next, we express condition (4.2.8) in terms of the dα -concentration (19) of u0 , as was for α = 2 in Proposition 4.1.6. Again, by scaling properties of system (4.1.4)–(4.1.5), it is sufficient to consider R = 1. Proposition 4.2.2 (Comparison of blowing up solutions). For d ≥ 3, α ∈ (0, 2), let the threshold number 𝒩 be the following: 𝒩 = inf {N : solution with the initial datum satisfying M(r) = N 1[1,∞) (r)

blows up in a finite time} .

Then the asymptotic relation 𝒩 ≲ σd d

α/2

holds as

d → ∞. α/2

Proof. Let us compute for the kernel Pt,α of the semigroup e−t(−Δ) using its representation with subordinator functions (34), the quantity Kα (d) = sup tPt,α ( t>0

s(α, d) )(0) |x|α

= s(α, d) sup t

1−d/α

t>0

∞

σd ∫ R( 0

r

t 1/α

)r −α+d−1 dr

∞

= s(α, d)σd ∫ R(ϱ)ϱd−1−α dϱ 0 ∞∞

(4.2.9) 2

= s(α, d)σd ∫ ∫ f1,α (λ)(4π)− /2 λ− /2 e−ϱ /4λ ϱd−α−1 dλ dϱ d

d

0 0

= s(α, d)σd π

−d/2

∞

∞

0

0

∫ f1,α (λ) ∫ 2−d e−τ λ− /2+ /2− /2 2d−α−1 τ d

d

α

d−α −1 2

dτ

4.2 Blowup à la Fujita of solutions of system with fractional diffusion

= 2α

Γ( d−α + 1)Γ(α) Γ( d−α ) 2 2

Γ( d2 − α + 1)Γ( α2 ) Γ( d2 )

= k0 (α) ≈ k(α)

+ 1) Γ( d−α ) Γ( d−α 2 2

Γ( d2 − α + 1) Γ( d2 )

∞

| 149

α

2−α ∫ f1,α (λ)λ− /2 dλ 0

,

for some constants k0 (α), k(α) > 0 independent of d, d → ∞, by formulas (1.3.2), (34). By the comparison principle in Chapter 2, Section 2.2.6 (and [52, Theorem 2.4]), if 0 ≤ u0 ≤ ϵuC for an ϵ ∈ [0, 1), then the solution is global, so it does not blow up in a finite time. Therefore, Kα (d) ≤ Cα (d) by this comparison result. So, now we need an upper estimate of the constant Cα (d) defined in (4.2.7). By definition (4.2.7), the global-in-time existence result in Chapter 2, Section 2.1 ([52, Theorem 2.4]) and relation (36), we obtain Kα (d) ≤ Cα (d) ≤

2d . d−2

(4.2.10)

Indeed, the left-hand side inequality is the consequence of the comparison principle in [52, Theorem 2.4]. Then, by representation (34), we have R󸀠 (ϱ) < 0 for ϱ > 0, and ∞

d

0 ≤ ϱR (ϱ) − R (ϱ) = ∫ f1,α (λ)(4πλ)− /2 ( 󸀠󸀠

󸀠

0

2 ϱ ϱ ϱ2 + )e−ϱ /4λ dλ, − 4λ2 2λ 2λ

(4.2.11)

so that d−1+ϱ

R󸀠󸀠 (ϱ) ≥ d − 2, |R󸀠 (ϱ)|

and the right-hand side inequality in estimate (4.2.10) follows. Now, we will test the normalized Lebesgue measure σd−1 dS on the unit sphere 𝕊d corresponding to the radial distribution function 1[1,∞) (r) using again the representation with subordinators (34) α/2

Lα (d) ≡ sup te−t(−Δ) (σd−1 dS) t>0

= sup t 1− /α R( d

t>0

1 ) t 1/α

= sup ϱd−α R(ϱ) ϱ>0

∞

2

= sup ∫ f1,α (λ)(4πλ)− /2 ϱd−α e−ϱ /4λ dλ ϱ>0

d

(4.2.12)

0

−α −d/2

=2 π

1−α/2

ϱ2 ϱ2 sup ∫ f1,α ( )( ) 4τ 4τ ϱ>0 ∞

0

τ

d−α −1 2

e−τ dτ.

150 | 4 Blowups à la Fujita From (4.2.12), the evident upper bound for Lα (d) is ∞

Lα (d) ≤ 2−α π − /2 sup f1,α (x)x1− /2 × ∫ τ d

α

x>0

d

− /2 ̃ = k(α)π Γ(

=

d−α −1 2

e−τ dτ

0

d−α ) 2

̃ ) Γ( d−α 2k(α) 2 σd Γ( d ) 2

1 −α/2 ̃ ≈ 2k(α) d σd

(4.2.13)

̃ for some constant k(α) > 0 independent of d, d → ∞, similarly as was in computations of (4.2.9), with the use of the Stirling formula (3.8.11). Now, we need an asymptotic lower bound for the quantity Lα (d). Observe that m ≡ max e−τ τ τ>0

d−α −1 2

d−α

= e−τ0 τ0 2

−1

with τ0 = ( d−α

d−α − 1) 2

2 d−α − 1) =e ( 2 d−α 1 ≈ Γ( ) 2 √π(d − α − 2)

− d−α +1 2

−1

(4.2.14)

1

holds by the Stirling formula (3.8.11). Now, let h ≍ d /2 . It is easy to check that d−α 1 min e−τ τ 2 −1 ≥ η m [τ0 ,τ0 +h]

for some η > 0, uniformly in d. Indeed, log

(d + h)d e−d−h h dh2 h2 h ) − h ≈ d − ). − h = 𝒪 ( = d log(1 + 2 d d 2d 2d dd e−d

From formulas (4.2.12) and (4.2.14), we infer Lα (d) ≥ π

−d/2

τ0 +1

sup ∫ f1,α ( ϱ>0

τ0

1−α/2

ϱ2 ϱ2 )( ) 4τ 4τ

τ

δh d−α ) Γ( √d 2 d α d ≈ π − /2 δΓ( )d− /2 . 2

d−α −1 2

e−τ dτ

d

≥ π − /2

(4.2.15)

Therefore, Lα (d) ≥ δ σ1 d− /2 holds. This, together with formula (4.2.13) leads to an estid mate of optimal order, and different from its counterpart for α = 2. Remark that if α

α/2

󵄩 󵄩 ℓα̃ (u0 ) ≡ sup t 󵄩󵄩󵄩e−t(−Δ) u0 󵄩󵄩󵄩∞ , t>0

(4.2.16)

4.2 Blowup à la Fujita of solutions of system with fractional diffusion | 151

then the comparison constants of ℓα̃ ( ⋅ ) with the dα -concentration || ⋅ || d depend on d. α

It is clear that if NLα (d) ≥ Cα (d), then N ≥ 𝒩 . Thus, if the radial distribution function M corresponding to the density u0 satisfies |u0|M d/α ≥ ||u0 || d ≥ r α−d M(r) > α

Cα (d) Lα (d)

for some r > 0,

then the solution with u0 as the initial condition blows up in a finite time, again by the comparison principle [52, Theorem 2.4]. Therefore, by inequality (4.2.10), we obtain α C (d) that 𝒩 = Lα (d) ≲ σd d /2 holds asymptotically as d → ∞. α

Remark 4.2.3 (Examples of blowing up solutions). Observe that for any initial condition u0 ≢ 0, there is N > 0 such that a sufficient blowup condition (4.2.8) is satisfied for Nu0 . Similarly as was in Remark 4.1.7, for α = 2, if u0 (x) = ηuC (x), then this sufficient 1 2d . Indeed, it condition for blowup (4.2.8) is satisfied for large d whenever η > k(α) d−2 C (d)

C (d)

1 2d 2 suffices to have η > Kα (d) , and by relation (4.2.10) asymptotically Kα (d) ≲ k(α) ≈ k(α) d−2 α α as d → ∞. More generally than in Remark 4.1.7, for each such η and sufficiently large R = R(η) > 1, the bounded initial condition of compact support u0 = η1{1≤|x|≤R} uC leads to a blowing up solution. It seems that this result cannot be obtained applying previous sufficient criteria for blowup, involving moments in [41] and in [52, Theorem 2.9].

5 Interpretations, complements, conjectures, et cetera 5.1 Dichotomy: local–global existence of solutions Let us recall the Cauchy problem studied in the preceding Chapters α

ut + (−Δ) /2 u + ∇ ⋅ (u∇v) = 0, Δv + u = 0,

x ∈ ℝd , t > 0, d

x ∈ ℝ , t > 0,

(5.1.1) (5.1.2)

supplemented with the initial condition u(x, 0) = u0 (x) ≥ 0.

(5.1.3)

Taking into account Theorem 1.4.1 in Chapter 1, Section 1.4 ([50, Theorem 2.1]), all the remarks on the critical Morrey space and the dα -radial concentration in previous Chapters, we formulate the following partial dichotomy result: Corollary 5.1.1. Let d ≥ 2, α ∈ (1, 2] and d + 1 > 2α. There exist two positive constants c(α, d) and C(α, d) such that i) if α ∈ (1, 2) and ||u0 || d < c(α, d), then problem (5.1.1)–(5.1.3) has a global-in-time α

solution; ii) ||u0 || d > C(α, d) implies that each nonnegative radially symmetric solution of probα

lem (5.1.1)–(5.1.3) blows up in a finite time.

This, together with estimate (4.2.13), shows that for α ∈ (1, 2), the discrepancy between bounds of the dα -radial concentration sufficient for either global-in-time exα istence or the finite time blowup, that is C(α,d) , is of order d /2 , similarly as was esc(α,d) tablished in [52, Remark 8.1] using an analysis of moments of solutions defined with C (d) compactly supported weight functions. Indeed, C(α, d) = Lα (d) , and c(α, d) ≥ ||uC || d = α

α

α

|uC|M d/α ≍ σd d /2−1 . In the case α = 2, Proposition 4.1.6, as we remarked after its proof, gives a better result: the discrepancy between the bounds of the d2 -radial concentration sufficient 1 for either the global-in-time existence or for the finite time blowup is of order d /2 . This improves the result in Chapter 3, Section 3.5 ([52, Remark 8.1]), where this quotient has been shown to be at most of order d. Remark 5.1.2. Observe that if u0 ≥ 0 and û0 ≥ 0, the condition 󵄩 󵄩 ℓ(u0 ) = sup t 󵄩󵄩󵄩etΔ u0 󵄩󵄩󵄩∞ ≫ 1 t>0

is sufficient for blowup. https://doi.org/10.1515/9783110599534-005

154 | 5 Interpretations, complements, conjectures, et cetera Indeed, ‖etΔ u0 ‖∞ = etΔ u0 (0) is evident by the representation 2

u(x, t) = ℱ −1 (e−t|ξ | û0 (ξ ))(x). More generally, observe that for nonnegative radial functions u0 the quantity supt>0 tetΔ u0 (0) is equivalent to supt>0 t‖etΔ u0 ‖∞ (even if etΔ u0 (0) is not equivalent to ‖etΔ u0 ‖∞ uniformly in t > 0). This can be shown using estimates for the Bessel kernel of the heat semigroup etΔ restricted to radial functions on ℝd in, for example, [117, Lemma 4.2]. The author is indebted to Jacek Dziubański and Marcin Preisner for interesting discussions on this issue.

5.2 Further comments on solvability of the classical and fractional Keller–Segel system 5.2.1 Initial traces Remark 5.2.1 (Initial trace of radial solutions). One can show that for d ≥ 3, similarly to the case α = 2, each nonnegative radial solution u on ℝd × (0, T) of system (5.1.1)– (5.1.2) has the initial trace, that is, u0 = lim u(t) t↘0

exists in the sense of weak convergence of measures. Moreover, u0 is uniformly in L1loc (ℝd ), and the solution u(t) satisfies the uniform bound in the local Morrey space d/α , that is, |u(t)|| d/α ≡ sup0 0 independent of R, since under the condition lim supR→0 Rα−d ∫{|x| Cα (d) we have wR (0) ≥ CRd−α . Thus, wR (T) > M and blowup occurs for T ≍ Rα when R → 0. Now, let us construct the data un that monotonically grow to such an initial condition u0 and lead to locally bounded solutions of the evolution problem (5.1.1)–(5.1.2) with their existence times shrinking to 0. Under assumptions (ii) of Theorem 3.6.1 ([52, Theorem 2.9]), we have Cα (d) < C = lim sup Rα−d ∫ u0 (x) dx R→0

≤ sup R R→0

{|x| 0, there exists arbitrarily small t∗ > 0 such that u is not in L∞ for 0 < t < t∗ . Remark 5.2.3. Note that, there is no nonnegative initial condition u0 with the Morrey space norm |u(t)||M d/α blowing up. Indeed, one can prove that each nonnegative local-

5.3 Optimal initial data for two dimensional Keller–Segel system

| 157

in-time solution of system (5.1.1)–(5.1.2) satisfies the condition lim sup r α−d

r→0, x∈ℝd

∫

󵄨 󵄨󵄨 󵄨󵄨u(y, t)󵄨󵄨󵄨 ≤ J(d, α) < ∞

{|y−x| 8π, nonnegative solutions blow up in a finite time, and for radially symmetric solutions, mass equal to 8π concentrates at the origin at the blowup time, see [50].

158 | 5 Interpretations, complements, conjectures, et cetera Our goal in this section is to give an alternative proof of the local-in-time existence of solutions to (5.3.1)–(5.3.3) when u0 ∈ ℳ(ℝ2 ) is a nonnegative finite measure with all its atoms of mass less than 8π. We believe that this approach is conceptually simpler than that in the recent paper [7] (which used elaborated arguments for interactions of solutions emanating from localized pieces of initial data), and those in previous papers [222], [223]. The latter approaches used heavily the free energy functional for system (5.3.1)–(5.3.2) considered in bounded domains. Moreover, our condition (5.3.5) seems to be more clear, and shows that measures with small atoms, which are not well separated as it was assumed in [7], are also admissible as initial data for the system (5.3.1)–(5.3.2). Here however, compared to [7], we obtain neither the uniqueness property of solutions nor the Lipschitz property of the solution map. The main result of this section is Theorem 5.3.1. Let 0 ≤ u0 ∈ L1 (ℝ2 ) ∩ L∞ (ℝ2 ) be a smooth initial density for (5.3.1)– (5.3.2) such that ‖u0 ∗ 1B(1) ‖∞ ≤ 8π − ε0

(5.3.5)

for some fixed ε0 > 0 and the unit ball B(1) centered at the origin of in ℝ2 . Then, there exists a solution of the problem (5.3.1)–(5.3.3) on the interval [0, t0 ] with t0 = t0 (ε0 , M) such that 󵄩 󵄩 sup t 1−1/p 󵄩󵄩󵄩u(t)󵄩󵄩󵄩p ≤ B,

0 0 and each solution u of (5.3.1)–(5.3.2) of mass M, the function uλ (x, t) = λ2 u(λx, λ2 t)

(5.3.7)

is also a solution, with its mass again equal to M. Of course, by a suitable scaling (5.3.7) of initial data, we see that we may satisfy the assumptions of the result on the local existence in Theorem 5.3.1 for any nonnegative u0 ∈ ℳ(ℝ2 ) with its atoms strictly less than 8π. Then, it is clear that we arrive at the following corollary (see also [7, Theorem 2]):

5.3 Optimal initial data for two dimensional Keller–Segel system

| 159

Corollary 5.3.2. The system (5.3.1)–(5.3.2) has a local-in-time solution for each initial nonnegative finite measure u0 with all its atoms strictly less than 8π. Indeed, it is sufficient to approximate (in the sense of the weak convergence of measures) such a measure u0 by a sequence of initial data satisfying (after the rescaling (5.3.7) with a single λ > 0) all the assumptions of Theorem 5.3.1. This approximation is possible by taking, for example, the approximative identity eδn Δ u0 for any sequence δn ↘ 0. Then, the existence time t0 is bounded from below by a positive quantity (since t0 depends on M and λ only). Next, we infer from the hypercontractivity estimate (5.3.6), and from the standard regularity theory for parabolic equations, that for every multiindex β 󵄩 󵄩󵄩 β 1/p−1−|β|/2 , 󵄩󵄩D u(t)󵄩󵄩󵄩p ≤ Cβ B t which permits us to pass to the limit with (a subsequence of) the approximating solutions, which are, in fact, smooth on ℝ2 × (0, t0 ). We obtain in such a way a solution to (5.3.1)–(5.3.2) with the measure u0 , and this solution is also smooth on ℝ2 × (0, t0 ). The proof of Theorem 5.3.1 will be a consequence of a well known fact in [16, 28] on the estimate of the existence time for a solution by mass of the initial condition only, see (5.3.10), by using a rather delicate argument of localization repeatedly. The existence of solutions results are proved (as in [16]) for the integral formulation of the system (5.3.1)–(5.3.3) already used in preceding Chapters u(t) = etΔ u0 + ℬ(u, u)(t),

(5.3.8)

whose solutions are mild solutions of the original Cauchy problem. Here, the bilinear term ℬ is defined as t

ℬ(u, z)(t) = − ∫(∇e

(t−s)Δ

) ⋅ (u(s) ∇(−Δ)−1 z(s)) ds.

(5.3.9)

0

It is well known that the heat semigroup etΔ satisfies the Lq − Lp estimates (39)–(40) for all 1 ≤ q ≤ p ≤ ∞. Moreover, for each p > 1 and z ∈ L1 (ℝ2 ), the following relation holds: 󵄩 󵄩 lim t 1−1/p 󵄩󵄩󵄩etΔ z 󵄩󵄩󵄩p = 0.

(5.3.10)

t→0

This is the consequence of, for instance, the following more general inequality valid for every finite measure μ ∈ ℳ(ℝ2 ) and every p > 1: 󵄩 󵄩 lim sup t 1−1/p 󵄩󵄩󵄩etΔ μ󵄩󵄩󵄩p ≤ Cp ‖μat ‖ℳ(ℝ2 ) ≡ Cp t→0

∑

󵄨 󵄨󵄨 󵄨󵄨μ({x})󵄨󵄨󵄨,

(5.3.11)

{x:μ({x})=0} ̸

where μat denotes the purely atomic part of the measure μ. The proof of rather subtle fact (5.3.11) is contained in [131, Lemma 4.4]. This fact, equivalent to the condition

160 | 5 Interpretations, complements, conjectures, et cetera (5.3.5), rescaled to other balls of a fixed radius, see (5.3.12) below, is crucial in the analysis of applicability of the Banach contraction argument to equation (5.3.8). We recall that the formulation of our existence results in [16] used, in fact, condition (5.3.11) in the definition of the functional space, where solutions have been looked for: 𝒳 ≡ {u : (0, T) → Lp (ℝ2 ) : ‖u‖𝒳 ≡ sup0 8π, which emanate from Mδ0 as (purely atomic) initial data. These are selfsimilar solutions, which are regular and nonunique for sufficiently large M, see Introduction, Section 1 and [3] and comments in [37, Remark 1] on the doubly parabolic Keller–Segel system with consumption terms. We will apply in the proof of Theorem 5.3.1 simple (but rather subtle) techniques of weight functions and scalings. The core of our analysis are the uniform (with respect to the initial distributions) estimates on the maximal existence time, expressed in terms of dispersion of the initial data. 5.3.1 Proof of Theorem 5.3.1 The proof of the estimate of the existence time in Theorem 5.3.1 is split into several lemmata. For any fixed x0 ∈ ℝ2 , we define the local moment of a solution u by Λ(t) ≡ ∫ ψ(x − x0 )u(x, t) dx.

(5.3.13)

Here, the weight function 2

ψ(x) = (1 − |x|2 )+ with

∇ψ(x) = −4x(1 − |x|2 )+ Δψ(x) = 16|x|2 − 8 ≥ −8

(5.3.14)

5.3 Optimal initial data for two dimensional Keller–Segel system

| 161

is a fixed radial, piecewise 𝒞 2 , nonnegative function ψ, supported on the unit ball such that ψ(0) = 1. Our particular choice of the function ψ here is, however, not critical. Lemma 5.3.4. Suppose that w = w(x) is a nonnegative function locally in L1 (ℝ2 ) ∩ L∞ (ℝ2 ), ∫B(1) w(x) dx ≤ m, and ϱ, δ ∈ (0, 1).

i)

Then there exists a number H0 ∈ (0, 1) such that ∫B(ϱ) w(x) dx ≤ (1 − δ)m implies

∫ ψ(x)w(x) dx ≤ (1 − H0 )m. ii) Similarly, there exists H1 ∈ (0, 1) such that if ∫B(1) w(x) dx ≤ m and ∫ ψ(x)w(x) dx ≥ (1 − H1 )m, then ∫B(ϱ) w(x) dx ≥ (1 − δ/2)m.

iii) Suppose that the inequality ∫ ψ(x)w(x) dx ≤ (1 − H)m holds with some H ∈ (0, 1). Then the bound ∫B(β) w(x) dx ≤ (1 − H/2)m holds for β2 ≤ H/4 ≤ 1/4. Proof. The properties (i–iii) are simple consequences of (5.3.14). Indeed for (i), ∫ ψ(x)w(x) dx ≤ ∫ w(x) dx + sup ψ(x) × B(1)\B(ϱ)

B(ϱ)

B(1)\B(ϱ)

2 2

≤ ∫ w(x) dx + (1 − ϱ ) B(ϱ)

w(x) dx

∫

∫

w(x) dx

B(1)\B(ϱ) 2

2

= (1 − ϱ2 ) ∫ w(x) dx + (1 − (1 − ϱ2 ) ) ∫ w(x) dx B(ϱ)

B(1) 2 2

2 2

≤ (1 − ϱ ) m + (1 − (1 − ϱ ) )(1 − δ)m

= (1 − H0 )m,

where 1 − H0 = (1 − ϱ2 )2 + (1 − (1 − ϱ2 )2 )(1 − δ) = 1 − δ(1 − (1 − ϱ2 )2 ). ii) is equivalent to (i) with δ replaced by δ/2. 1−H iii) For |x| ≤ β, β2 ≤ H/4 and H ≤ 1, the inequality ψ(x) ≥ 1− H/2 holds. Next, we show a result on the dispersion of the initial data evolving according to (5.3.1)–(5.3.2). Lemma 5.3.5. Let u be a solution to (5.3.1)–(5.3.2) such that for t ∈ [0, A] ‖u0 ∗ 1B(R0 ) ‖∞ ≤ m

(5.3.15)

for some A > 0, R0 = 6 ⋅ 128 πM > 0 and m0 ≤ m ≤ 8π − ε0 . Then, there exist numbers ε 0

A1 = A1 (M, ε0 ), δ = δ(M, ε0 , m0 ), and ϱ = ϱ(M, ε0 , m0 ), such that if ∫|y−x |≤ϱ u(y, t) dy ≥ 0 (1 − δ)m for some t ∈ [0, A], then the differential inequality Λ󸀠 (t) ≤ −ϑ holds with some ϑ = ϑ(M, ε0 , m0 ) > 0.

162 | 5 Interpretations, complements, conjectures, et cetera Proof. First we give a uniform estimate of the time derivative of the moment Λ(t) 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨Λ (t)󵄨󵄨 ≤ CM . Let us compute the time derivative of Λ using equations (5.3.1)–(5.3.2) and (5.3.14). Symmetrizing the bilinear integral ∫ u(x, t)∇v(x, t) ⋅ ∇ψ(x) dx with the solution v of 1 (5.3.2) given by v(x, t) = − 2π ∫ u(y, t) log |x − y| dy, we obtain Λ󸀠 (t) = ∫ u(x, t)Δψ(x) dx +

(5.3.16)

∇ψ(x) − ∇ψ(y) 1 ⋅ (x − y)u(x, t)u(y, t) dx dy. ∬ 4π |x − y|2

From (5.3.16) and (5.3.14), we immediately get Λ󸀠 (t) ≤ 8M + 4M 2 . Using (5.3.14), the bound |∇ψ(x) − ∇ψ(y)| ≤ 4 and the relation ∫B(ϱ) u(x, t) dx ≤ 8π 1 R0 −ϱ

with ϱ ≤ 1 < 2 ≤ R0 and

󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ 󵄨󵄨

≤

2 , R0

∫

|x| 0 is any small positive number and α can be chosen as α = α0 σ 2 for some α0 > 0. Theorem 5.3.10 has been proved (even for sign changing measures) in [16, Theorem 2] (see also [28, proof of Theorem 2.2]) using a standard contraction argument applied to the formulation (5.3.8). Lemma 5.3.12. Suppose that u0 ∈ L1 (ℝ2 ) ∩ L∞ (ℝ2 ) is a smooth nonnegative function satisfying the condition ‖u0 ∗ ψ‖∞ ≤ 8π − ε0 . Then, the solution u with the initial condition (5.3.3) u0 exists at least on the time interval ε [0, 2C0 ]. M

ε

Proof. The inequality Λ󸀠 (t) ≤ CM implies that Λ(t) ≤ 8π − ε0/2 for t ≤ τ1 ≡ min{ 2C0 , τ}, M where τ is the maximal existence time of u. By iii) of Lemma 5.3.4, we obtain the bound ‖u(t) ∗ 1B(ϱ1 ) ‖∞ ≤ 8π − ε0/4 for all t ≤ τ1 . From Corollary 5.3.9, we infer that there exists 1

σ0 = ητ1/2 such that ‖u(τ2 ) ∗ 1B(σ0 ) ‖∞ ≤ m0 for τ1/2 < τ2 < τ1 . By Theorem 5.3.10, the solution with u0 = u(x, τ2 ) as the initial condition exists for t ∈ [0, ατ2 ] with some α > 0 independent of τ2 . Therefore, by Theorem 5.3.10, this solution can be continued onto 1 the interval [0, τ1 + ατ2 ]. This solution satisfies the estimate ‖u(τ1 + ατ2 )‖p ≤ Cτ2/p−1 for each p ∈ [4/3, 2]. Finally, a recurrence argument permits us to obtain a classical solution u = u(x, t) on the whole interval [0, T0 ] with T0 = T0 (ε0 , M), and applying once more Corollaries 5.3.8 and 5.3.11, satisfies the hypercontractive estimate ‖u(t)‖p ≤ 1 Ct /p−1 for p ∈ [4/3, 2]. The extrapolation of that estimate to the whole range of p ∈ (1, ∞) is standard, see [28].

5.4 Blowup of solutions to a general diffusive aggregation model The nonexistence of global-in-time solutions is studied using classical moment method for a class of aggregation equations, involving Lévy diffusion operators and

166 | 5 Interpretations, complements, conjectures, et cetera general, not necessarily scale invariant as is the Newtonian one, interaction kernels. This section is based on the paper [42], and the presented approach offers an alternative way to establish blowups than the kinetic method in many papers on aggregation equations either without diffusion or with weak (degenerate, singular, et cetera) diffusion, see [11, 12, 170] and [176–178, 251]. Further studies on aggregation and concentration (when ν → 0) phenomena for models (5.4.1) with singular kernels are in [150, 151], [179] (α ∈ (0, 2)), and [24] when α = 2, see [73] for the Burgers equation. We consider in this section the Cauchy problem for the evolution equation α

ut + ν(−Δ) /2 u + ∇ ⋅ (u∇K ∗ u) = 0,

u(x, 0) = u0 (x),

(5.4.1) (5.4.2)

which describes swarming, collective motion, and aggregation phenomena in biology and mechanics of continuous media. Here x ∈ ℝd , t ≥ 0, and u = u(x, t) ≥ 0 refer to either the population density of a species or the density of particles in a granular media. When ν = 0, equation (5.4.1) can be considered as either a conservation law with a nonlocal (quadratic) nonlinearity or a transport equation with nonlocal velocity, and its character depends strongly on the properties of the kernel K. A classical choice for K is K(x) = e−|x| or, more generally, K is a radially symmetric function of r = |x|. Nonincreasing kernels correspond to the attraction of particles, whereas nondecreasing ones are repulsive. Local and global existence of solutions to the inviscid equation (5.4.1) (ν = 0) has been thoroughly studied in [170] under some additional hypotheses on the kernel, see also [11, 13]. In particular, kernels that are smooth (not singular) at the origin (x = 0) lead to the global-in-time existence of solutions, see [13, 170]. Mildly singular kernels (for example, 𝒞 1 off the origin, such as K(x) = e−|x| ) may lead to blowup of solutions either in finite or infinite time [11–13, 170, 176–178]. Strongly singular kernels, such as potential-type (arising in chemotaxis theory, see [18, 41]) K(x) = c|x|β−d ,

(5.4.3)

with 1 < β < d (so, in particular, the Newtonian potential kernel K(x) = cd |x|2−d , d ≥ 3), usually lead to finite time blowup of “large” solutions, see [41, 62, 178]. Equations (5.4.1) with fractional diffusion term (ν > 0) have been introduced in the physical literature and studied in, for example, [34, 62, 63, 121], starting in the nineties of the 20th century. The linear term in (5.4.1) is described by a fractional power of the Laplacian operator in ℝd (or, more generally, by a Lévy diffusion operator) defined in the Fourier variables by α/2

α

̂ (ξ ), ℱ ((−Δ) u)(ξ ) = |ξ | u with 0 < α ≤ 2.

(5.4.4)

5.4 Blowup of solutions to a general diffusive aggregation model | 167

When ν > 0 and K is a radially symmetric and nonincreasing function of r = |x| with a mild singularity at r = 0, equation (5.4.1) then features a diffusive term, which spreads the distribution of particles and a nonlinear drift term, which concentrates it, thus acting in the opposite direction. The fundamental question for the Cauchy problem (5.4.1)–(5.4.2) is to decide whether u, governed by the competition between the nonlinear transport term and the linear dissipative term, can describe aggregation phenomena or not. Of course, the answer may depend on the (regularity and) size of initial data. Typical approaches to prove a finite time aggregation include an extension of the method of characteristics [12, 177], the energy method (for example, [11, 13, 176, 177]), and the moment (or virial) method. The latter has been first applied to mean field models for self-gravitating particles and chemotaxis systems, [15], and recently in [18, 41]. At this point, we mention that the characteristics method obviously cannot be applied in the presence of diffusion. Our aim in this section is to present a simple virial-type argument showing finite time blowup of a large class of solutions of (5.4.1)–(5.4.2) if ν ≥ 0 and 0 < α < 1. The results we obtained are similar to those in [177], but we believe that our proofs are more direct and simpler. In addition, our assumptions on the initial data (see (5.4.7) and (5.4.8)) and the kernel (see (5.4.5) and (5.4.6)) are less restrictive than those in [176] (radial symmetry and high localization on the initial data, the kernel K being a nonincreasing function of r = |x|), and [177] (existence of exponential moments or even compact support assumption and K(x) = e−|x| ). In particular, the kernel K is allowed to have a repulsive part. The case of the strong dissipation 1 < α ≤ 2 and nonlinearities with potential kernels (5.4.3) has been considered in [41, 62, 178], where threshold conditions on the values of α, β, d have been determined so that solutions can be either continued indefinitely in time, or they can blow up in a finite time for suitable initial data. But, for weakly singular kernels as the ones considered in this section, the strong dissipation 1 < α ≤ 2 prevents finite time blowup and global solutions exist, see [121] and [176, Theorem 3]. We do not consider here local-in-time existence of solutions, their positivity, and mass conservation properties since these topics have been discussed in detail in, for example, [11, 41, 62, 170]. One should note that the existence theory in [176, 177] is developed in the spirit of arguments used for conservation laws in [170], that is, without taking into account regularization effects of the diffusion, whereas [41, 62, 63, 121] employed those effects in a significant way. Our results employ a crucial property of the gradient ∇K(x) of the convolution x kernel in (5.4.1), namely, the fact that − |x| is its homogeneous part near the origin. More precisely, we will use two sets of assumptions on the kernel K. There is a locally Lipschitz continuous function k such that K(x) = k(|x|) for x ∈ ℝd , and

168 | 5 Interpretations, complements, conjectures, et cetera (H1) either: there is K0 > 0 such that − K0 ≤ k 󸀠 ≤ 0

and

κR = − sup k 󸀠 > 0 (0,R)

(5.4.5)

for each R > 0; (H2) or: k 󸀠 (r) = −k1󸀠 (r) + k2󸀠 (r) for r > 0, and there are K0 > 0, K1 > 0, K2 ≥ 0, and δ ∈ [0, 1) such that 󵄨󵄨 󸀠 󵄨󵄨 δ 󵄨󵄨k (r)󵄨󵄨 ≤ K0 (1 + r ) ,

K1 ≤ k1󸀠 (r) ,

k2󸀠 (r) ≤ K2 r δ

(5.4.6)

for all r > 0. We will consider solutions, which are even in x, which is implied by the assumption that the initial condition u0 is even: x ∈ ℝd ,

u0 (x) = u0 (−x),

(5.4.7)

together with the radial symmetry of the kernel K and the uniqueness of solutions to (5.4.1)–(5.4.2). Moreover, we need that M1 = ∫ |x| u0 (x) dx < ∞.

(5.4.8)

ℝd

As we have already remarked, the total mass is conserved during the evolution of (5.4.1)–(5.4.2) ∫ u(x, t) dx = M = ∫ u0 (x) dx. ℝd

(5.4.9)

ℝd

Now we are ready to state main results of this work. Theorem 5.4.1. Assume that ν = 0. Consider a nonnegative and integrable initial condition u0 ≢ 0 satisfying (5.4.7) and (5.4.8). i) If k fulfills (5.4.5) or (5.4.6) with K2 = 0, then the solution u to the Cauchy problem (5.4.1)–(5.4.2) ceases to exist in a finite time. ii) If k fulfills (5.4.6) and M1 is sufficiently small, then the solution u to the Cauchy problem (5.4.1)–(5.4.2) ceases to exist in a finite time. Let us emphasize here that the assumptions (5.4.5) and (5.4.6) with K2 = 0 apply to two different classes of kernels k: indeed, k 󸀠 is required to be bounded in the former, but can vanish at infinity (in the sense that κR might decay to zero as R → ∞). The growth condition is less restrictive for the latter, but k 󸀠 is not allowed to vanish at infinity. We also point out here that we do not know whether the smallness of M1 is a necessary condition for finite time blowup to occur when k fulfills (5.4.6).

5.4 Blowup of solutions to a general diffusive aggregation model | 169

It follows from Theorem 5.4.1 that, in the absence of diffusion, and if the kernel is attractive (k 󸀠 ≤ 0), finite time blowup takes place for any nonzero initial data, whereas a partially attractive kernel seems to require the initial data to be sufficiently concentrated for this phenomenon to occur. Since diffusion is expected to act also as a repulsive term, localization of the initial data seems also to be needed for finite time blowup when ν > 0, even if the kernel is attractive. Indeed, we have the following result: Theorem 5.4.2. Consider a nonnegative and integrable initial condition u0 ≢ 0 satisfying (5.4.7) and (5.4.8). Assume that ν > 0, 0 < α < 1, and that k fulfills either (5.4.5) or (5.4.6). If M is sufficiently large and M1 is sufficiently small, then the solution u to the Cauchy problem (5.4.1)–(5.4.2) ceases to exist in a finite time. Observe that, besides localization of the initial data as in the partially repulsive case, Theorem 5.4.2 also requires the total mass M to be sufficiently large. This is due to the fact that, in the proof, it does not seem possible to balance the contribution from the diffusion with that from the drift term. In contrast to [176] and [177, Theorem 12], our conditions on u0 , guaranteeing finite time blowup, do not require the L1 -norm of u0 to be smaller (in a suitable sense) than that of u0 (K ∗ u0 ). We also improve [177, Theorem 8], where u0 is assumed to be in L1 (ℝd ; e2|x| dx). 5.4.1 Virial inequalities For γ ∈ (0, 1] and x ∈ ℝd , we define Wγ (x) = wγ (|x|) with wγ (r) =

1 ((1 + r)γ − 1), γ

r ≥ 0.

Evidently, Wγ ≥ 0 is a Lipschitz continuous function, which will be used as a weight function. We list below some properties of Wγ and wγ that we will repeatedly use in the remainder of this section. Lemma 5.4.3. Consider γ ∈ (0, 1) and α ∈ (γ, 1). Then (−Δ) /2 Wγ ∈ L∞ (ℝd ). α

Proof. Recall that, for x ∈ ℝd , the Lévy–Khintchine representation formula reads α

−(−Δ) /2 Wγ (x) = 𝒜 lim δ↘0

∫ {|y|>δ}

Wγ (x + y) − Wγ (x) |y|d+α

dy,

with 𝒜 defined in (24), and alternatively either [115, Theorem 1] or [147]. Given x ∈ ℝd , we set ϱ = ϱ(|x|) = max {1, |x|}, and use the monotonicity and subadditivity of the function r 󳨃→ r γ to obtain 1 󵄨󵄨 (1 + |x| + |y|)γ − (1 + |x|)γ α/2 󵄨 1 dy 󵄨󵄨(−Δ) Wγ (x)󵄨󵄨󵄨 ≤ ∫ 𝒜 γ |y|d+α ℝd

170 | 5 Interpretations, complements, conjectures, et cetera

≤

1 γ−1 ∫ γ(1 + |x|) |y|1−d−α dy γ B(0,ϱ)

+

1 γ

(1 + |x|)γ + |y|γ − (1 + |x|)γ dy |y|d+α

∫ ℝd \B(0,ϱ)

γ−1

≤ C(d) (1 + |x|)

≤ C(d, α, γ) ϱγ−α ,

ϱ1−α ϱγ−α + C(d) 1−α γ(α − γ)

and the right-hand side of the above inequality is bounded since α > γ. Additional properties of the weight wγ are summarized in Lemma 5.4.4. Consider γ ∈ (0, 1]. For each ε > 0, there exists a constant Cε > 0 such that the inequalities (1 − wγ󸀠 (r)) ≤ (1 − γ) wγ (r) and

wγ (r) ≤

1 γ r ≤ ε + Cε wγ (r) γ

(5.4.10)

hold for all r ≥ 0. For δ ∈ [0, γ) and R > 1, we have rδ ≤

2wγ (r) Rγ−δ

for

r ≥ R.

(5.4.11)

Proof. The first inequality in (5.4.10) follows from the observation that the function f (r) = (1 − γ)((1 + r)γ − 1) − γ + γ(1 + r)γ−1 satisfies f 󸀠 (r) = γ(1 − γ)(1 + r)γ−2 r ≥ 0 and f (0) = 0. The second inequality in (5.4.10) is clear for small r ≥ 0 and suitably large C = Cε , as well as for large r ≫ 1. Finally, if R > 1 and r ≥ R, we have 1+r

wγ (r) = ∫ sγ−1 ds ≥ r (1 + r)γ−1 ≥ 1

(1 + r)γ r γ−δ δ ≥ r , 2 2

from which (5.4.11) readily follows since γ > δ. Next we derive an identity, involving the moment Iγ of a nonnegative solution u of (5.4.1) defined by Iγ (t) = ∫ Wγ (x)u(x, t) dx ℝd

whenever it is meaningful, for example, if u ∈ L1 ((0, T); (1 + |x|) dx).

(5.4.12)

5.4 Blowup of solutions to a general diffusive aggregation model | 171

Lemma 5.4.5. For each t ≥ 0, we have dIγ dt

(t) = −ν𝒟(t) + 𝒜1 (t) + 𝒜2 (t),

(5.4.13)

where α/2

𝒟(t) = ∫ u(x, t)[(−Δ) Wγ ](x) dx, ℝd

𝒜1 (t) = ∬(wγ (|x|) − 1)k (|x − y|) 󸀠

󸀠

𝒜2 (t) = ∬ k (|x − y|) 󸀠

x x−y ⋅ u(x, t) u(y, t) dx dy, |x| |x − y|

x x−y ⋅ u(x, t) u(y, t) dx dy. |x| |x − y|

Proof. The evolution of Iγ is governed by (5.4.1) so that dIγ dt

α

(t) = −ν ∫ Wγ (x)[(−Δ) /2 u](x, t) dx ℝd

+ ∫ wγ󸀠 (|x|) ℝd

x u(x, t) ⋅ (∇K ∗ u)(x, t) dx |x| α

= −ν ∫ u(x, t)[(−Δ) /2 Wγ ](x) dx ℝd

+ ∬ wγ󸀠 (|x|)k 󸀠 (|x − y|)

x x−y ⋅ u(x, t) u(y, t) dx dy |x| |x − y|

= −ν𝒟(t) + ∬(wγ󸀠 (|x|) − 1)k 󸀠 (|x − y|) + ∬ k 󸀠 (|x − y|)

x x−y ⋅ u(x, t) u(y, t) dx dy |x| |x − y|

x x−y ⋅ u(x, t) u(y, t) dx dy, |x| |x − y|

whence inequality (5.4.13). The next step is to find suitable upper bounds for 𝒟, 𝒜1 , and 𝒜2 . Such an estimate for 𝒟 follows at once from Lemma 5.4.3 and conservation of mass property (5.4.9), and reads as follows: α/2 󵄩 󵄩 𝒟(t) ≤ 󵄩󵄩󵄩(−Δ) Wγ 󵄩󵄩󵄩∞ ∫ u(x, t) dx, 𝒟(t) ≤ C(d, γ, α)M

ℝd

provided α ∈ (γ, 1).

(5.4.14)

Now we turn to 𝒜1 and 𝒜2 , and first consider the case, where k satisfies (5.4.5). Lemma 5.4.6. Assume that k fulfills (5.4.5). Then, for any R > 1, 𝒜1 (t) ≤ (1 − γ) M K0 Iγ (t) ,

(5.4.15)

𝒜2 (t) ≤ M κ2R (

(5.4.16)

Iγ (t)

wγ (R)

−

M ). 2

172 | 5 Interpretations, complements, conjectures, et cetera Proof. We infer from (5.4.5), (5.4.9), and the first inequality in (5.4.10) that 𝒜1 (t) ≤ ∬(1 − wγ (|x|)) 󵄨󵄨󵄨k (|x − y|)󵄨󵄨󵄨 u(x, t) u(y, t) dx dy

󵄨

󸀠

󵄨

󸀠

≤ (1 − γ) K0 ∬ wγ (|x|) u(x, t) u(y, t) dx dy , whence (5.4.15). Symmetrizing the double integral 𝒜2 , we obtain 𝒜2 (t) =

=

1 y x−y x − )⋅ u(x, t) u(y, t) dx dy ∬ k 󸀠 (|x − y|) ( 2 |x| |y| |x − y|

x y |x| + |y| 1 (1 − ⋅ ) u(x, t) u(y, t) dx dy . ∬ k 󸀠 (|x − y|) 2 |x − y| |x| |y|

Since k 󸀠 is nonpositive by (5.4.5) and 1≤

(x, y) ∈ ℝd × ℝd ,

|x| + |y| , |x − y|

(5.4.17)

we deduce from (5.4.5) and (5.4.9) that, for any R > 1, 𝒜2 (t) ≤

≤

x y 1 ⋅ ) u(x, t) u(y, t) dx dy ∬ k 󸀠 (|x − y|) (1 − 2 |x| |y| 1 2

≤− ≤−

∫ k 󸀠 (|x − y|) (1 −

∫

B(0,R) B(0,R)

κ2R 2

∫

∫ (1 −

B(0,R) B(0,R)

x y ⋅ ) u(x, t) u(y, t) dx dy |x| |y|

x y ⋅ ) u(x, t) u(y, t) dx dy |x| |y|

κ2R x y ⋅ ) u(x, t) u(y, t) dx dy ∬(1 − 2 |x| |y|

+ κ2R ∫ (

∫

ℝd ℝd \B(0,R)

(1 −

x y ⋅ ) u(y, t) dy)u(x, t) dx |x| |y| 2

κ2R x (M 2 − (∫ u(x, t) dx) ) 2 |x| wγ (|y|) u(y, t) dy)u(x, t) dx + κ2R ∫ ( ∫ wγ (R)

≤−

ℝd ℝd \B(0,R)

≤−

κ2R 2 κ M + 2R M Iγ (t) , 2 wγ (R)

since the evenness of the function x 󳨃→ u(x, t) warrants that ∫ ℝd

x u(x, t) dx = 0 |x|

The proof of Lemma 5.4.6 is then complete.

for t ≥ 0 .

(5.4.18)

5.4 Blowup of solutions to a general diffusive aggregation model | 173

We now derive the counterpart of Lemma 5.4.6 when the kernel k satisfies the weaker assumption (5.4.6). Though the proof roughly proceeds along the same steps as that of Lemma 5.4.6, it is more complicated because some terms, involving k1 and k2 have to be handled separately. Lemma 5.4.7. Assume that k fulfills (5.4.6). Then, for any R > 1, γ ∈ (δ, 1), and ε ∈ (0, 1), there is a constant Cε󸀠 depending only on ε, γ and δ such that δ

δ−γ

𝒜1 (t) ≤ K0 [(1 − γ) (2M + M R + Iγ (t)) + 2 M R 𝒜2 (t) ≤ M(

] Iγ (t) ,

K1 K + K2 Cε󸀠 )Iγ (t) − ( 1 − 2 K2 ε)M 2 . wγ (R) 2

(5.4.19) (5.4.20)

Proof. On the one hand, we infer from assumption (5.4.6) and inequalities (5.4.10) that, for R > 1, 𝒜1 (t) ≤ ∬(1 − wγ (|x|)) 󵄨󵄨󵄨k (|x − y|)󵄨󵄨󵄨 u(x, t) u(y, t) dx dy

󵄨

󸀠

󵄨

󸀠

≤ K0 ∬(1 − wγ󸀠 (|x|)) (1 + |x − y|δ ) u(x, t) u(y, t) dx dy ≤ (1 − γ) K0 ∬ wγ (|x|) (1 + |y|δ ) u(x, t) u(y, t) dx dy + K0

∫ (1 − wγ󸀠 (|x|)) |x|δ u(x, t)( ∫ u(y, t) dy) dx B(0,R)

+ K0

ℝd

(1 − wγ󸀠 (|x|)) |x|δ u(x, t)( ∫ u(y, t) dy) dx .

∫ ℝd \B(0,R)

ℝd

We next use (5.4.9), (5.4.10), (5.4.11), and the property wγ󸀠 ≤ 1 to obtain γ

𝒜1 (t) ≤ (1 − γ) K0 ( ∫ (1 + (1 + |y|) ) u(y, t) dy) Iγ (t) ℝd

∫ wγ (|x|) |x|δ u(x, t) dx

+ (1 − γ) M K0 2MK + γ−δ 0 R

B(0,R)

∫

wγ (|x|) u(x, t) dx

ℝd \B(0,R)

≤ (1 − γ) K0 (2 M + γ Iγ (t)) Iγ (t) + (1 − γ) K0 Rδ M Iγ (t)

2 M K0 Iγ (t) Rγ−δ ≤ K0 [(1 − γ) (2M + M Rδ + Iγ (t)) + 2 M Rδ−γ ] Iγ (t) , +

hence (5.4.19) follows.

174 | 5 Interpretations, complements, conjectures, et cetera On the other hand, after the symmetrization of the double integral in 𝒜2 , it follows from the positivity of k1󸀠 in (5.4.6) that, for R > 1, 𝒜2 (t) =

= =

1 y x−y x − )⋅ u(x, t) u(y, t) dx dy ∬ k 󸀠 (|x − y|) ( 2 |x| |y| |x − y|

1 x y |x| + |y| (1 − ⋅ ) u(x, t) u(y, t) dx dy ∬ k 󸀠 (|x − y|) 2 |x − y| |x| |y| 1 2

B(0,R) B(0,R)

+

+

≤−

∫ k 󸀠 (|x − y|)

∫ 1 2

∫ (

k 󸀠 (|x − y|)

∫

B(0,R) ℝd \B(0,R)

1 2

1 2

x y |x| + |y| (1 − ⋅ ) u(x, t) u(y, t) dx dy |x − y| |x| |y|

( ∫ k 󸀠 (|x − y|)

∫

ℝd \B(0,R) ℝd

∫ k1󸀠 (|x − y|)

∫

B(0,R) B(0,R)

x y |x| + |y| (1 − ⋅ ) u(y, t) dy)u(x, t) dx |x − y| |x| |y|

x y |x| + |y| (1 − ⋅ ) u(y, t) dy)u(x, t) dx |x − y| |x| |y|

x y |x| + |y| (1 − ⋅ ) u(x, t) u(y, t) dx dy |x − y| |x| |y|

1 x y |x| + |y| + ∬ k2󸀠 (|x − y|) (1 − ⋅ ) u(x, t) u(y, t) dx dy . 2 |x − y| |x| |y| Recalling (5.4.6), (5.4.9), (5.4.17), and (5.4.18), we proceed as in the proof of Lemma 5.4.6 to conclude that ∫ k1󸀠 (|x − y|)

∫

B(0,R) B(0,R)

≥ K1

∫

∫ (1 −

B(0,R) B(0,R)

≥ K1 M 2 − 2 K1 ∫ ( ℝd

≥ K1 M 2 − 2

x y |x| + |y| (1 − ⋅ ) u(x, t) u(y, t) dx dy |x − y| |x| |y|

x y ⋅ ) u(x, t) u(y, t) dx dy |x| |y| ∫

u(y, t) dy)u(x, t) dx

ℝd \B(0,R)

M K1 I (t) . wγ (R) γ

Using once more assumption (5.4.6), property (5.4.9), and the obvious bound 0≤

x y x y x−y |x| + |y| (1 − ⋅ )=( − )⋅ ≤ 2, |x − y| |x| |y| |x| |y| |x − y|

(x, y) ∈ ℝd × ℝd ,

we find 1 x y |x| + |y| (1 − ⋅ ) u(x, t) u(y, t) dx dy ∬ k2󸀠 (|x − y|) 2 |x − y| |x| |y| ≤ K2 ∬ |x − y|δ u(x, t) u(y, t) dx dy

5.4 Blowup of solutions to a general diffusive aggregation model | 175

≤ K2 ∬(|x|δ + |y|δ ) u(x, t) u(y, t) dx dy ≤ 2 M K2 ∫ |x|δ u(x, t) dx ≤ 2 M K2 (ε M +

δ γ−δ ( ) γ γε

(γ−δ)/δ

Iγ (t)) ,

whence (5.4.20). 5.4.2 Finite time blowup 5.4.2.1 The inviscid case ν = 0 Now we are ready to prove the first blowup result for the inviscid model (5.4.1)–(5.4.2) with ν = 0, and begin with the case of a nonincreasing kernel K. Proof of Theorem 5.4.1 i). We argue by contradiction and assume the solution u of (5.4.1)–(5.4.2) to be well defined for all times. Combining (5.4.13), (5.4.15), and (5.4.16), we end up with dIγ dt

(t) ≤ Λγ,R (Iγ (t)) = M [(1 − γ) K0 +

κ2R κ M2 ] Iγ (t) − 2R wγ (R) 2

for all R > 1 and γ ∈ (0, 1). Since Λγ,R is a nondecreasing function, we realize that we have Iγ (t) ≤ Iγ (0) + Λγ,R (Iγ (0)) t for t ≥ 0 as soon as Λγ,R (Iγ (0)) < 0. Then, of course, Iγ attains zero at some finite time t0 , which is impossible for nonnegative regular solutions to (5.4.1)–(5.4.2), a contradiction with the global existence. We next observe that we can always find γ ∈ (1/2, 1) and R > 1 such that Λγ,R (Iγ (0)) < 0, or equivalently [(1 − γ) K0 +

κ M κ2R ] I (0) < 2R . wγ (R) γ 2

Indeed, if γ ∈ (1/2, 1), we have wγ (r) ≤ r for r ≥ 0 and wγ (r) ≥ √r/2 for r ≥ 1. Therefore, choosing R > 1 such that M1 < (M √R)/8, and then γ ∈ (1/2, 1) such that (1 − γ) K0 M1 < (κ2R M)/4, we realize that [(1 − γ) K0 +

κ M κ2R 2κ ] I (0) ≤ [(1 − γ) K0 + 2R ] M1 < 2R . √R wγ (R) γ 2

With this choice of R and γ, we have Λγ,R (Iγ (0)) < 0, and the proof is complete. Proof of Theorem 5.4.1 ii). We again argue by contradiction, and assume the solution u of (5.4.1)–(5.4.2) to be well defined for all times. Combining (5.4.13), (5.4.19), and (5.4.20), we end up with dIγ dt

(t) ≤ Λγ,R,ε (Iγ (t)) for t ≥ 0 ,

(5.4.21)

176 | 5 Interpretations, complements, conjectures, et cetera where Λγ,R,ε (z) = M [(1 − γ) K0 (2 + Rδ ) + + (1 − γ) K0 z 2 − (

2K0 K1 + + K2 Cε󸀠 ]z γ−δ wγ (R) R

K1 − 2 K2 ε)M 2 2

for all R > 1 and ε ∈ (0, 1). As before, the inequality (5.4.21) contradicts the global existence of nonnegative regular solutions to (5.4.1)–(5.4.2) as soon as Λγ,R,ε (Iγ (0)) < 0. Since Λγ,R,ε is an increasing function and Iγ (0) ≤ M1 , we have Λγ,R,ε (Iγ (0)) ≤ Λγ,R,ε (M1 ). Observing that an appropriate choice of ε (sufficiently small) and R (sufficiently large) warrants Λγ,R,ε (0) < 0, we thus have Λγ,R,ε (M1 ) < 0 provided M1 is small enough. Hence, for such a choice of ε and R, finite time blowup of the solution to (5.4.1)–(5.4.2) occurs as claimed. Finally, if K2 = 0, we may argue as at the end of the proof of Theorem 5.4.1(i) to show that, given any nonzero initial condition u0 , we may find R large enough and γ close to one such that Λγ,R,ε (Iγ (0)) < 0, which completes the proof. Clearly, the only term in (5.4.21) that prevents Theorem 5.4.1(ii) from being valid for an arbitrary nonzero initial condition u0 is the term K2 Cε󸀠 Iγ , which cannot be made arbitrarily small by an appropriate choice of γ, R, and ε. This term reflects the deviation of k from being decreasing, and thus the partially repulsive behavior of k. 5.4.3 The dissipative case ν > 0 The second result applies to solutions with suitably large initial data in the dissipative case. Proof of Theorem 5.4.2. Assume first that k fulfills assumption (5.4.5). We argue by contradiction and assume the solution u of problem (5.4.1)–(5.4.2) to be well defined for all times t ≥ 0. Combining (5.4.13), (5.4.14), (5.4.15), and (5.4.16), we end up with dIγ dt

(t) ≤ Λγ,R (Iγ (t)) = M [(1 − γ) K0 +

κ2R ] I (t) wγ (R) γ

+ νC(d, γ, α) M −

κ2R M 2 2

for all R > 1 and γ ∈ (0, α). As before, the above inequality contradicts the global existence of nonnegative regular solutions to (5.4.1)–(5.4.2) as soon as Λγ,R (Iγ (0)) < 0, the latter being true if Λγ,R (M1 ) < 0. Fix γ ∈ (0, α) and R > 1, and assume that M > 4ν C(d, γ, α)/κ2R . Then, Λγ,R (0) ≤ −Mν C(d, γ, α) < 0 so that Λγ,R (M1 ) < 0 if M1 is sufficiently small. If k fulfills assumption (5.4.6), the proof is similar and relies on (5.4.13), (5.4.14), (5.4.19), and (5.4.20).

5.5 Hypercontractivity of the linearization around the singular solution | 177

In contrast to the proof of Theorem 5.4.1(i), we cannot play with the parameter γ in the proof of Theorem 5.4.2 when k fulfills (5.4.5). Indeed, γ is limited by the constraint γ < α, and cannot be chosen arbitrarily close to one. This explains the necessity of having sufficiently localized initial data, in the sense that M1 is required to be small enough.

5.5 Hypercontractivity of the linearization around the singular solution We begin with a study of the linearized evolution operator around the singular Chandrasekhar solution for the Cauchy problem (5.1.1)–(5.1.3). A prerequisite to these considerations is an analysis of diffusion operators with singular potentials, see [71, 143, 211, 212]. Theorem 5.5.1. Assume that d ≥ 15. Suppose that u(x, t) is a global-in-time radial solution satisfying 0 ≤ u(x, t) ≤ uC (x) for all x ∈ ℝd and t ≥ 0. There exists 1 < p0 < 2 such that if uC − u0 ∈ L2 (ℝd ) ∩ Lp (ℝd ) with some p ∈ (p0 , 2), then and

󵄩 󵄩󵄩 󵄩󵄩uC − u(t)󵄩󵄩󵄩2 ≤ ‖uC − u0 ‖2

(5.5.1)

󵄩 󵄩󵄩 −d(1−1) 󵄩󵄩uC − u(t)󵄩󵄩󵄩2 ≤ C(p, d)t 2 p 2 ‖uC − u0 ‖p

(5.5.2)

for all t > 0 and a number C(p, d) independent of t and u. We leave some problems to be solved, including the following: Existence of solutions with such 0 ≤ u0 (x) ≤ uC (x) data; Strategy of the proof: The first step is the approximation of the initial condition u0 = u0k + z0 by (1 − 1/k)uC + ⋅ ⋅ ⋅, with ‖z0 ‖L∞ ≪ 1, ‖z0 ‖q ≪ 1, q > d/2 so that ||uk || ≤ (1 − 1/k)2σd by Theorem 2.1.1. p Next, energy bound ‖∇z /2 ‖∞ ≪ 1 is to be proved. The sign of w should be controlled. To prove Theorem 5.5.1, we consider the linearization of problem (5.1.1)–(5.1.3). We substitute w(x, t) = uC (x) − u(x, t), ΔφC + uC = 0 to (5.1.1) to get wt = Δw − ∇ ⋅ (uC ∇φ) − ∇ ⋅ (w∇φC ) + ∇ ⋅ (w∇φ),

Δφ + w = 0,

w(x, 0) = w0 (x).

(5.5.3) (5.5.4) (5.5.5)

Let us denote the linear differential operator ℒw = −Δw + ∇uC ⋅ ∇φ − uC w + ∇ ⋅ (w∇φC ).

(5.5.6)

In the following, we study properties of the operator ℒ, see an analogous approach in [211].

178 | 5 Interpretations, complements, conjectures, et cetera Lemma 5.5.2. Assume that d ≥ 15. Then there exists a constant λ > 0 such that the operator ℒ defined in (5.5.6) satisfies the following inequality: ⟨ℒw, w⟩ ≥ λ‖∇w‖22 ,

(5.5.7)

for all radial functions w ∈ H 1 (ℝd ). Proof. Integrating by parts leads to ⟨ℒw, w⟩ = ⟨−Δw + ∇uC ⋅ ∇φ − uC w + ∇ ⋅ (w∇φC ), w⟩ = ‖∇w‖22 + ∫ ∇uC ⋅ ∇φw dx − ∫ uC w2 dx − ∫ w∇φC ⋅ ∇w dx =

ℝd

ℝd

ℝd

‖∇w‖22

2

+ ∫ ∇uC ⋅ ∇φ w dx − ∫ uC w dx − ℝd

ℝd

1 ∫ ∇w2 ⋅ ∇φC dx. 2 ℝd

We integrate again by parts; use the fact that ΔφC = −uC and compute the gradient of the Chandrasekhar solution uC to have ⟨ℒw, w⟩ = ‖∇w‖22 + ∫ ∇uC ⋅ ∇φ w dx −

3 ∫ uC w2 dx 2 ℝd

ℝd

= ‖∇w‖22 − 4(d − 2) ∫ ℝd

w2 x ⋅ ∇φ w dx − 3(d − 2) dx. ∫ |x|4 |x|2

(5.5.8)

ℝd

Using the Hardy inequality (2.1.29), we get ⟨ℒw, w⟩ ≥ (1 −

x 12 )‖∇w‖22 − 4(d − 2) ∫ 4 ⋅ ∇φ w dx. d−2 |x| ℝd

To deal with the last integral, involving radial functions, we apply Lemma 2.1.8: −4(d − 2) ∫ ℝd

4(d − 2) x ⋅ ∇φ w dx = ∫ |x|−d−2 w ∫ w(y) dy dx σd |x|4 =4

ℝd ∞ −d−2

d−2 ∫r σd

|y|≤|x|

Mr M dr

0

∞

= −2

d−2 ∫ (r −d−2 )r M 2 dr σd 0

=2

∞

(d − 2)(d + 2) ∫ r −d−3 M 2 dr ≥ 0 σd 0

irrespective of the signs of w and M(r, t) = ∫{|x| 0 is satisfied if d ≥ 15.

Lemma 5.5.3. Assume that d ≥ 15. There exists 1 < p0 < 2 such that if w ∈ L2 (ℝd ) ∩ Lp (ℝd ) with some p ∈ (p0 , 2) is a nonnegative radial function, then there exists λ > 0 such that 󵄩 p 󵄩2 ⟨ℒw, wp−1 ⟩ ≥ λ󵄩󵄩󵄩∇w /2 󵄩󵄩󵄩2 .

(5.5.9)

Proof. We integrate by parts and evaluate the gradient of the Chandrasekhar solution uC to get ⟨ℒw, wp−1 ⟩ =

x 4(p − 1) 󵄩󵄩 p/2 󵄩󵄩2 p−1 󵄩∇w 󵄩󵄩2 − 4(d − 2) ∫ 4 ⋅ ∇φ w dx |x| p2 󵄩 ℝd

− ∫ uC wp dx − (p − 1) ∫ wp−1 ∇w ⋅ ∇φC dx ℝd

ℝd

=

4(p − 1) 󵄩󵄩 p/2 󵄩󵄩2 x p−1 󵄩∇w 󵄩󵄩2 − 4(d − 2) ∫ 4 ⋅ ∇φ w dx |x| p2 󵄩 ℝd

p−1 − ∫ uC w dx − ∫ ∇wp ⋅ ∇φC dx p p

ℝd

ℝd

4(p − 1) 󵄩󵄩 p/2 󵄩󵄩2 x p−1 = 󵄩󵄩∇w 󵄩󵄩2 − 4(d − 2) ∫ 4 ⋅ ∇φ w dx 2 |x| p ℝd

2p − 1 − ∫ uC wp dx p ℝd

=

x 4(p − 1) 󵄩󵄩 p/2 󵄩󵄩2 p−1 󵄩∇w 󵄩󵄩2 − 4(d − 2) ∫ 4 ⋅ ∇φ w dx |x| p2 󵄩 p

−

ℝd

2(2p − 1)(d − 2) w ∫ 2 dx. p |x| ℝd

Now, we use the Hardy inequality (2.1.29) together with Lemma (2.1.8) to obtain 4(p − 1) 8(2p − 1) 󵄩󵄩 p/2 󵄩󵄩2 − )󵄩∇w 󵄩󵄩2 p(d − 2) 󵄩 p2 4(d − 2) + ∫ |x|−d−2 wp−1 ∫ w(y) dy dx σd

⟨ℒw, wp−1 ⟩ ≥ (

ℝd

|y|≤|x|

180 | 5 Interpretations, complements, conjectures, et cetera

≥(

4(p − 1) 8(2p − 1) 󵄩󵄩 p/2 󵄩󵄩2 )󵄩∇w 󵄩󵄩2 , − p(d − 2) 󵄩 p2

since 4(d − 2) ∫ |x|−d−2 wp−1 ∫ w(y) dy dx ≥ 0. σd ℝd

|y|≤|x|

This completes the proof of Lemma 5.5.3, since for d ≥ 15 and p ∈ (p0 , 2) the number 4(p−1) − 8(2p−1) is positive. p(d−2) p2 5.5.1 An alternative approach to the property of hypercontractivity of the operator ℒ This approach works for d ≥ 17 only, but needs neither assumption on the sign of w nor radial symmetry of w. Now, let us estimate the middle term in formula (5.5.8) 󵄨󵄨 󵄨󵄨 |∇φ| |w| x 󵄨󵄨 󵄨 dx 󵄨󵄨4(d − 2) ∫ 4 ⋅ ∇φ w dx 󵄨󵄨󵄨 ≤ 4(d − 2) ∫ 󵄨󵄨 󵄨󵄨 |x| |x|2 |x| 2 4 ‖∇w‖22 ≤ 4(d − 2) d(d − 4) d − 2 by the Hardy inequality (2.1.29) for w and the Rellich inequality for ∇φ 2 d2 (d − 4)2 󵄩󵄩󵄩󵄩 f 󵄩󵄩󵄩󵄩 2 󵄩󵄩 2 󵄩󵄩 ≤ ‖Δf ‖2 , 󵄩󵄩 |x| 󵄩󵄩2 16

(5.5.10)

32 12 see [5, (6), (6.2.3)]. To conclude, observe that d(d−4) < 1 − d−2 for d ≥ 17. Thus, we obtain the following property: ℒ is the generator of a holomorphic semigroup.

Proof of Theorem 5.5.1. We multiply equation (5.5.3) by wp−1 and use Lemma 5.5.3 to have 1 1 d 󵄩󵄩 󵄩p p+1 p−1 󵄩w(t)󵄩󵄩󵄩p = −⟨ℒw, w ⟩ − ( + 1) ∫ w dx ≤ 0, p dt 󵄩 p ℝd

since w ≥ 0. Hence 󵄩 󵄩󵄩 󵄩󵄩w(t)󵄩󵄩󵄩p ≤ ‖w0 ‖p

for p ∈ (p0 , 2).

We multiply equation (5.5.3) by w and use Lemma 5.5.2 to get 1 d 󵄩󵄩 3 󵄩2 3 2 󵄩w(t)󵄩󵄩󵄩2 = ⟨−ℒw, w⟩ − ∫ w dx ≤ −⟨ℒw, w⟩ ≤ −λ‖∇w‖2 , 2 dt 󵄩 2 ℝd

(5.5.11)

5.6 Classical diffusion case: super- and subsolutions | 181

since w ≥ 0. Using the Gagliardo–Nirenberg inequality 4p 2+ d(2−p)

‖w‖2

4p

≤ C(d, p)‖∇w‖22 ‖w‖pd(2−p)

for

1 < p < 2,

we obtain 2+s

‖w(t)‖2 1 d 󵄩󵄩 󵄩2 󵄩w(t)󵄩󵄩󵄩2 ≤ −C 2 dt 󵄩 ‖w‖sp with s = form

4p d(2−p)

and C > 0. Applying (5.5.11) leads to the differential inequality of the 1 󸀠 1+s/2 f (t) ≤ −C‖w0 ‖−s (t) p f 2

for the function f (t) = ‖w(t)‖22 and C > 0, which gives the algebraic decay of the L2 norm 󵄩 󵄩󵄩 −d(1−1) 󵄩󵄩w(t)󵄩󵄩󵄩2 ≤ Ct 2 p 2 ‖w0 ‖p , with the exponent depending on p ∈ (p0 , 2). We also need a perturbation result with convergence of a radial solution u(t) to uC as t → ∞ obtained via the hypercontractivity property of the semigroup linearized at uC in high dimensions d ≥ 15 and d ≥ 17, respectively, which is not proved yet.

5.6 Classical diffusion case: super- and subsolutions This section which has a character of a preliminary study, is an excerpt from an unpublished manuscript written in collaboration with Ignacio Guerra, developing ideas in [39, 55, 20]. Related constructions have been used in [227, 247]. For comparison principles, we refer the reader to [9, Lemma 5.1], [253, Proposition 2.4] and [47]. These families of super- and subsolutions are important in analysis of local and global existence of solutions as well as for determination of profiles of solutions at the blowup time. For d ≥ 2, it is easy to check that for radially symmetric solution u the mass distribution function M = M(r, t), r = |x|, satisfies a simple nonlinear parabolic equation d−1 1 𝜕M = Mrr − Mr + r 1−d MMr , 𝜕t r σd

(5.6.1)

supplemented with the initial condition M(r, 0) = M0 (r),

(5.6.2)

182 | 5 Interpretations, complements, conjectures, et cetera see [39, (6)]. Equation (5.6.1) has singular coefficients, so that even the existence of solutions issues are not trivial, see [43, 44] and [20] in the case d = 2. Here, due to the interpretation of M as a distribution function, we suppose that M(., t) is a nondecreaŝ ∈ [0, ∞]. ing function, and M(0, t) = 0, M(∞, t) = M Introducing a new independent variable y = r d and denoting Q(y, t) = M(r, t), we obtain a slightly simpler equation 𝜕Q d 2 = d2 y2− /d Qyy + QQy , 𝜕t σd

(5.6.3)

together with the conditions Q(0, t) = 0,

̂ Q(∞, t) = M,

Q(y, 0) = Q0 (y),

(5.6.4)

a nonuniformly parabolic problem. Here, Q is also a nondecreasing function. A standard approach to solve problem (5.6.3)–(5.6.4) is to regularize equation (5.6.3) replacing this by d 𝜕Q 2 = d2 (y + ε)2− /d Qyy + QQy , 𝜕t σd

(5.6.5)

with ε > 0, and pass to the limit ε ↘ 0, see [166] and [39]. Solutions of the initial-boundary value for equation (3.1.8) satisfying the bound M(r, 0) ≤ Cr d−2 for some C > 0 can be constructed by a regularization of equation (3.1.8) (r 󳨃→ (r + ε), ε > 0) and crucial is the use of a supersolution (an upper barrier) C(r + ε)d−2 . Of course, Q(y) = 2σd (y + ε)1− /d 2

(5.6.6)

is an obvious stationary solution. First, solutions Q = Qε of problem (5.6.5), (5.6.4), with regular (Q0 )ε ∈ H 1 (0, Y) for any finite Y > 0 instead of ∞, exist on (0, Y) × (0, T0 ) for each T0 < ∞. This, together with a standard application of the parabolic regularity theory in [166], leading to a locally uniform Schauder-type bound with any β ∈ (0, 1), η > 0, τ > 0, ‖Qε ‖

2+β,1+β/2

𝒞s,t

([η,Y]×[τ,T0 ])

≤ C(β, η, τ, T0 ),

(5.6.7)

with C = C(β, η, τ, T0 ) independent of ε > 0, are consequences of [166, Theorem III.10.1, IV.10.1]. Whenever approximate solutions can be controlled in a vicinity of the origin (so that Qε (y, t) → 0 as y → 0 to guarantee the boundary condition expressing that no mass concentrates at the origin at any time t > 0), we may pass with Qε to the limit ε ↘ 0. Thus, we obtain—by Ascoli–Arzelà theorem applied, thanks to (5.6.7)—a limit function Q, which solves problem (5.6.3)–(5.6.4) with Q0 . Q is a classical solution

5.6 Classical diffusion case: super- and subsolutions | 183

in (0, ∞) × (0, T0 ). Detailed presentations of those approximation procedures can be found in [39] as well as (for d = 2) in [43, 44]. Here, we omit details, and only give a construction of suitable local-in-time supersolutions for problem (5.6.4)–(5.6.5), which show that there exists a limit Q satisfying Q(0, t) = 0 and Qy (0, t) < ∞ for 0 < t < T with some T > 0. 5.6.1 Local-in-time supersolutions Supersolutions constructed below are for d ≥ 3, for d = 2 see [20]. Denote by ℒε the (semilinear) operator 2 𝜕w d 𝜕w 2 𝜕 w − d2 (y + ε)2− /d 2 − w , 𝜕t σd 𝜕y 𝜕y

ℒε w =

(5.6.8)

with the obvious meaning for ℒ = ℒ0 ; for each ε > 0, the comparison principle for ℒε obviously holds, see [166]. Define the function Qm (y, t) =

C(y + ε) , (y + ε) + k(T − t)

(5.6.9)

where C > 0 is any positive constant and k, T are suitable positive constants (in general, k ≥ 1, T ≪ 1) to be determined later. Elementary calculations lead to 3

((y + ε) + k(T − t)) ℒε Qm = C(y + ε)(k((y + ε) + k(T − t)) + 2d2 y1− /d k(T − t) − 2

≥ C(y + ε) k(T − t)(k −

d Ck(T − t)) σd d C). σd

Clearly, for each C > 0, there is k > 0 such that ℒε Qm ≥ 0 so that Qm is a supersolution of the regularized problem (5.6.4)–(5.6.5) if, of course, initially Q0 (y) ≤

Cy C(y + ε) ≤ = Qm (y, 0). y + kT (y + ε) + kT

This condition is satisfied for many initial conditions Q0 if a sufficiently small T > 0 is chosen, for each ε > 0. Note that the total mass of Qm is finite: limy→∞ Qm (y, t) = C. Another class of supersolutions, this time with infinite mass, but for initial conditions Q0 with slightly another behavior in the vicinity of y = 0, is constructed defining the function Qq (y, t) =

A(y + ε) , (y + ε)2/d + k(T − t)

(5.6.10)

184 | 5 Interpretations, complements, conjectures, et cetera where A > 0 is a positive constant and k, T are suitable positive constants (in general; k ≥ 1, T ≪ 1) to be determined later. Again, elementary calculations lead to 3

2

2

((y + ε) /d + k(T − t)) ℒε Qq = A(y + ε)(k((y + ε) /d + k(T − t)) 2

+ 2(d − 2)(y + ε) /d + 2(d + 2)k(T − t) −A

d d−2 2 (y + ε) /d + k(T − t))). ( σd d

It is clear that if we choose k ≥ Qq satisfying

Q0 (y) ≤

d A (and a fortiori k σd

≥

d−2 A), then ℒε Qq σd

≥ 0. Therefore,

A(y + ε) Ay ≤ = Qq (y, 0) y2/d + kT (y + ε)2/d + kT

is a supersolution of the regularized problem (5.6.4)–(5.6.5). Again, there are many initial conditions Q0 that satisfy the above condition for suitably small T > 0. For instance, those u0 ∈ L∞ with u0 (x) ≍ |x|−2 , x → ∞ (and this for all d ≥ 3, even for d = 3 when uC is neither weak nor distributional solution of (5.1.1) in the sense of Section 2) verify that condition above. Both families of supersolutions will be modified to construct subsolutions with similar behavior at y = 0 as Qm in (5.6.9) and Qq in (5.6.10), respectively, but with completely different blowup properties when t ↗ T. 5.6.2 Blowing up subsolutions Recall that the solution (either on a ball or on ℝd ) ̃ t) = Q(y,

y

2/d

4σd y + 2(d − 2)(T − t)

(5.6.11)

is an explicit blowing up solution (4.1.21), see also [80, 65]. This has been obtained in Chapter 4, Section 4.1 as a kind of minimal rate blowing up solution. Just as for d = 2 in [20], we construct for d ≥ 3 blowing up subsolutions for (5.6.3) on ℝd . Let Cy + δy, y + k(T − t)3

(5.6.12)

Ay + δy. y2/d + k(T − t)2

(5.6.13)

Qm (y, t) = and Qq (y, t) =

5.6 Classical diffusion case: super- and subsolutions | 185

Proposition 5.6.1. Here, for each C > 0, there exist positive k, T, δ such that Qm is a

subsolution of (5.6.3). Similarly, for each A > 2 d+2 σ , there exist positive k, T, δ such d d that Qq is a subsolution of (5.6.3). The terms δy in (5.6.12) and (5.6.13) are useful in having Qm and Qq as subsolutions. 2

However, they could be modified for large y > 0 to have limy→∞ y /d−1 Qq (y, t) < ∞, and limy→∞ Qm (y, t) < ∞. For instance, the implosion wave on the ball in [80] has been first constructed for d = 3 using matched asymptotic expansions by M. Herrero, E. Medina, and J. J. L. Velázquez in [135]. This leads to a blowup with mass concentration. To check the subsolution property for Qm , let us calculate 3

(y + k(T − t)3 ) ℒQm = Cy(3k(T − t)2 (y + k(T − t)3 ) + 2d2 y1− /d k(T − t)3 − 2

−

(5.6.14)

d d 2 Ck(T − t)3 − δ(y + k(T − t)3 ) σd σd

d d 3 δ(y + k(T − t)3 )k(T − t)3 − δ2 C −1 (y + k(T − t)3 ) ). σd σd

Now, we estimate the positive terms on the right-hand side of identity (5.6.14) above. Using the Hölder inequality, we get for the second term d(2dy

1−2/d

2/d

d/(d−2)

d/2

2 d d − 2 4σd −1 ) ≤ d(( ( ( C ) C) ) + d 4σd d d ≤

y)

d 1 d 1 ( C + c1 C −d/(d−2) y) ≤ ( C + δy) σd 2 σd 2 4σ

for some constant c1 (= 2 d ( dd )d/(d−2) ), depending on d only, and if δ = c1 C −d/(d−2) . The first term can be bounded by (d−2)σ

3k(T − t)2 (y + k(T − t)3 ) ≤

1 d C(k(T − t)3 ) 2 σd

2/3

3 d 1 1 3 C) ) (y + k(T − t)3 ) + ( 1/3 ( 3 3k 4 σd

≤ 1

−3

d 1 3 ( Ck(T − t)3 + δ2 C −1 (y + k(T − t)3 ) ) σd 2

4σ

1

if δ = c2 C /2 k /2 , with c2 = dd depending on d only. This yields ℒQm ≤ 0. Similarly, we have for a subsolution of the second type 3

(y /d + k(T − t)2 ) ℒQq = Ay(2k(T − t)(y /d + k(T − t)2 ) 2

+ 2(d − 2)y /d + 2(d + 2)k(T − t)2 − 2

2

d − 2 2/d d Ay − Ak(T − t)2 σd σd

186 | 5 Interpretations, complements, conjectures, et cetera

− − For

A σd

d d − 2 2/d 2/d 2 2 δ(y /d + k(T − t)2 ) − δy (y + k(T − t)2 ) σd σd

(5.6.15)

d 3 2 2 δk(T − t)2 (y /d + k(T − t)2 ) − δ2 A−1 (y /d + k(T − t)2 ) ). σd

> 2, let us estimate the first term on the right-hand side above: > 2 d+2 d k 2/d 2 (y + k(T − t)2 ) . ν

2k(T − t)(y /d + k(T − t)2 ) ≤ νk(T − t)2 + 2

Then, for ν > 0 small enough so that 2(d + 2) + ν ≤ σd A, and with δ satisfying σd δ ≥ kν , d d all the positive terms on the right-hand side of (5.6.15) are absorbed by negative ones, leading to ℒQq ≤ 0. Of course, when t ↗ T, the subsolution Qm (⋅, t) loses its boundary condition at y = 0 jumping to C > 0, so that the corresponding density u(⋅, t) approaches a pointmeasure singularity Cδ0 at the origin: lim lim Qm (y, t) = C.

y→0 t↗T

Now it is clear that the subscript m signifies measure. Similarly, for t ↗ T, the subsolution Qq (⋅, t) loses its derivative and resembles Ay1− /d with A big enough near y = 0, that is, its profile is similar to the distribution of 1 the singular solution uC in the original variables r = |x| = y /d (for which A = 2σd ): 2

2

lim lim y /d−1 Qq (y, t) = A.

y→0 t↗T

The subscript q signifies here quenching for equation (5.6.3), as is for uC , that is, the density and the derivative of the integrated density at the origin become infinite for solutions of equation (5.6.3). Note that Qq (y, t) ≪ Qm (y, t) for any choice of A, C, k, T, and each sufficiently small y > 0. Surely, in each of the above cases, the blowup can occur earlier than t = T. It is of interest to note that for each A > 0 and each B > 0, the function Q(y) =

Ay y+B

is a subsolution of (5.6.3) on the finite interval (0, ( 2dA2 σ )d/(d−2) ). Indeed, d

(y + B)3 ℒQ(y) = ABdy(2d2 y1− /d − 2

A ). σd

A simpler candidate for Qq is Qq (y, t) =

Ay , y + φ(t) 2/d

(5.6.16)

5.6 Classical diffusion case: super- and subsolutions | 187

where φ = φ(t) is a strictly decreasing function on [0, T), with either T < ∞ or T = ∞, and with the bounded derivative φ󸀠 (t): limt→T φ(t) = 0. This is a blowing-up subsolution for a suitable choice of parameters A, φ. Let us calculate φ󸀠 (t) 𝜕 Qq (y, t) = −Ay 2/d , 𝜕t (y + φ(t))2 A 𝜕 2 2 Q (y, t) = 2/d ((1 − )y /d + φ(t)), 𝜕y q d (y + φ(t))2

2 2 Ay /d−1 2 𝜕2 ((1 − )y /d + (1 + )φ(t)). Q (y, t) = − 2 d d 𝜕y2 q (y /d + φ(t))3 2

Next, we have 2

3

2

(y /d + φ(t)) ℒQq = Ay(−φ󸀠 (t))(y /d + φ(t)) + A(d − 2)y1+ /d (2 − 2

+ Ayφ(t)(2(d + 2) −

A ) σd

dA ). σd

Now, it is clear that for A > 2 d+2 σ , and |φ󸀠 (t)| ≤ dA − 2(d + 2), the function Qq is a d d σd subsolution, which blows up (not later than) T—finite or infinite. In particular, φ(t) = k(T − t)d with suitably small k or φ(t) = e−ct satisfy these restrictions. 5.6.3 Another family of blowing up subsolutions They keep the finite value of the density at the origin, that is, with u(0, t) = 0 for any t < T: Qq (y, t) =

Ay1+γ y + (T − t)δ

(5.6.17)

with γ ∈ (0, 2/d), δ ∈ (1, d/2) and suitably large A. Other useful special barriers (on the unit ball only, see [220]): Qq (y, t) =

Ay yβ + k(T − t)

(5.6.18)

with 2/d ≤ β < 1, σA ≥ 2dβ (≥ 4). If k ≤ d2 β(1 − β), then Qq is a subsolution. d With respect to those examples, the following Question can be posed: What happens if β = 2/d and 2 < σA < 4, and we start the evolution with such a d subsolution Qq ? Do they blow up? Problem. This construction is not valid for β = 1 (subsolutions with jumping boundary condition at y = 0 when t = T). How to modify this? In another way, see of course (5.6.12).

188 | 5 Interpretations, complements, conjectures, et cetera

5.7 Miscellanea, questions and conjectures, research plans There is a couple of open questions related to the constructions in the preceding sections, for instance: Does there exist an initial condition 0 ≤ u0 ∈ L1 (ℝd ) with u blowing up like a point measure cδ? The construction in [135] is based on formal asymptotic expansions. Remark 5.7.1. Concerning selfsimilar solutions mentioned in Remark 2.1.10, it is an interesting problem of characterization of all radial selfsimilar solutions of the parabolic-elliptic Keller–Segel model in higher dimensions (as it has been done for the doubly parabolic Keller–Segel model in two dimensions in [3]). A direct approach to prove that these solutions exist for each ε ∈ (0, 1) with the initial data u0 = εuC follows the strategy in [259]: First, prove the existence of solutions of the form (2.1.30) for small ε > 0 (this is a particular case of Proposition 1.4.2(ii)). Next, derive a priori estimates of solutions to equation for selfsimilar solutions in Lq , q > 1, independent of ε ∈ (0, 1). Finally, apply the Leray–Schauder method. This is the scheme used for proving the existence of selfsimilar solutions of the Navier–Stokes system with either classical ([157]) or fractional diffusion ([168]). Another approach to radially symmetric selfsimilar solutions uses monotonicity properties of solutions to equation (2.1.8) with the form (2.1.30) (or equation (2.9.6)), and is described in a work in progress [47]. One may also apply the dynamical systems approach sketched in Chapter 2, Section 2.9. Remark 5.7.2. There is an extensive literature on the time asymptotics of solutions at single blowup points at the blowup time. In particular, these points are classified according to the rates of blowup: type I—with a rate intrinsic to the scaling of the model, that is, (T − t)−1 , and type II—with a faster rate, see Remark 4.1.2. See [132, 188, 187, 189, 190, 215, 152] for more detail. Remark 5.7.3. For related models of aggregation of particles under diffusions and interactions of different type, we refer the reader to [151, 150, 12, 42, 229, 23]. Remark 5.7.4. The local attracting property of the stationary solution Mb reported in Introduction, Section 1 is a rather weak property. In particular, this does not give any information on the asymptotic behavior of solutions starting from initial data, such as M0 (s) = 8πs/(s + 2 + cos s) satisfying the relation M3 ≤ M0 ≤ M1 , but (M0 − Mb ) ∉ L1 (0, ∞) for any b > 0. For such an initial condition, a possible conjecture is that M will oscillate between M3 and M1 throughout time evolution. We may thus expect the long time behavior of solutions to be extremely complicated in the critical case. For recent results on this asymptotics described by global attractors, see [199, 180, 181]. The authors of [153] discuss the nonlinear nonlocal parabolic equation vt = Δv + arising in some scaling limits of doubly parabolic chemotaxis systems.

v λ ∫ eev dx Ω

5.7 Miscellanea, questions and conjectures, research plans | 189

For solutions u with the initial condition u0 of compact support, the moment W(t) = ∫ u(x, t)ψγ (x, t) dx, where ψγ (x, t) = e(T−t)Δ (|x|−γ ), 0 < γ < d can be used. This would permit detecting singularities of u(., T) of the type |x|γ−d as G in Chapter 4 does for u with limt↗T ‖u(t)‖∞ = ∞. Besides either concentrations of mass or emergence of singularities resembling singular solutions uC , other mechanisms of blowup are possible. In particular, blowup of certain modes of the Fourier transform of solutions has been shown in [25], similarly to an analogous phenomenon for a simplified model of Navier–Stokes equations discovered in [192].

5.7.1 Nonlinear heat equation with quadratic nonlinearity and chemotaxis It is clear that (6) with p = 2, and the diffusion term νΔu, ν ≥ 0, ut − νΔu = u2 ,

(5.7.1)

ut − νΔu = u2 − ∇u ⋅ ∇v

(5.7.2)

and (1.1.1)–(1.1.2) written as

differ by the term ∇u⋅∇v, which has dissipative properties. According to previous analyses, this term prevails over u2 in the one dimensional case, since the one dimensional chemotaxis possesses exclusively global-in-time solutions, see [21, 23, 61] for more precise comparisons. Indeed, with u = −vxx = wx , we have wt = νwxx + wwx , so that the Burgers equation with all global-in-time solutions appears. In the two dimensional case, this term can be again dominant, but only if ‖u0 ‖1 ≤ 8πν. In higher dimensions, this occurs if ‖u0 ‖M d/2 ≤ 2σd ν. This analogy extends to the inviscid case ν = 0 when for d = 1 we have inviscid Burgers equation with shocks, similarly to the one dimensional ordinary differential equation ut = u2 . 5.7.2 Heuristics To finish, let us summarize, a hierarchy of size of blowing up solutions (either on a ball or in ℝd ): Let us take the initial condition u0 smooth and u0 ≈ εuC (off x = 0) and consider the Chandrasekhar solution uC (x) = 2(d − 2)|x|−2 . A list of results includes the following: ε < 1 then u global and smooth ε = 1—stationary, singular ]—open problem: local versus global existence ε ∈ (1, d+2 d 2 —the derivative blows up at y = 0 like y− /d (by a subsolution) ε > d+2 d

190 | 5 Interpretations, complements, conjectures, et cetera ε = 2—the example from [80], see (4.1.21) ε = dβ, 2/d ≤ β < 1—a more singular blowup of the derivative at y = 0 ε > d/2—from the Feller test for linear diffusion-convection equations: the solution looses the blowup condition at y = 0 ε > d—blowup using moments with power-like weights, [39] 2 ε > 2(d − 1)—blowup such that y2− /d Qy (y, t) does not tend to 0 as y → 0 (on the 2/d−1 looses the condition Q(0, t) = 0. ball), see [56], but then Q ≍ y 1 2 ε > 4 e d (≈ 1.81d) (asymptotically as d → ∞)—no weak solutions at all (“an instantaneous blowup”), see [52] and [51] for a twice worse estimate. ε > 2dσd —“easy” proof of blowup, see [39].

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Index aggregation equation 165, 166 asymptotic relation – f ≈ g XXI – f ≍ g XXI – f ≲ g XXI Banach contraction theorem 2, 4, 12, 57, 61, 160 barrier 24, 37, 38, 45 Bessel kernel 115, 119–121, 123 Beta function 37, 134 blowing up solution – explicit example 142, 144, 151 blowup XV, 1, 73, 76 – in finite time XIV, 73 – in infinite time XVI – in terms of concentration of data 116 – instantaneous 190 – interior and boundary points 87 – of Fourier transform 189 – of N-symmetric solutions – with concentration of mass 88 – of radial solutions – with large concentration 87, 104 – rates XIX, 141 – sufficient condition 85, 108, 109, 135 – time estimate 91 – with mass concentration 185, 188, 189 Bochner subordination formula XXII boundary condition – Neumann XII – no-flux XII Brownian diffusion XXIII, 19, 75, 120 Burgers equation XVI, 23, 166 Chandrasekhar solution XII, 6–8, 32, 34, 145, 155, 189 comparison of concentration and Morrey norm 35, 55, 87, 116, 135 comparison of moment and concentration 108 comparison of moment and Lebesgue norm 79 comparison of moment and Morrey norm 75 comparison principle 19, 33 – averaged 34, 43, 143 – for radial distribution function 24 – pointwise 34, 36, 54 concavity constant 112, 113, 120–122

concentration – d2 -radial XXI

– dα -radial XX, 6, 22, 135, 153 – ψ-radial 86 – of mass XV, 85 – vanishing estimate 22 concentration of mass 31, 74, 86–88 – quantized 86, 89 conservation of mass XII, 1, 22, 29, 30, 32, 66, 77, 79, 86, 117 continuation of solution past blowup 86 convexity of weight 82 convolution of powers of |x| 8 critical mass XV critical quantity XVIII, 2, 73, 86, 104 critical space 2, 35, 116 critical value of mass 58 cross-diffusion XI, XIII decay property 53 dichotomy XVIII, 145, 153 discontinuity with respect to initial data 155, 156 discrepancy 136, 153 Duhamel formula 1, 3, 9, 14, 50, 56, 59, 61, 159 energy method 167 entropy XIII – relative XVI equation for radial distribution function XVI, 23, 67, 143, 181 equivalent conditions for blowup XVIII, 144 estimate – hypercontractive 124, 127, 128 – interpolation 4, 56 – Lq − Lp for heat semigroup XXIII, 60 – Lq decay 22 – of heat semigroup XXIV – in Morrey spaces XXIV – in Morrey spaces with two indices XXIV – of heat semigroup on measures 159 – of Riesz potential 4 – in Morrey spaces 56 – of the gradient of heat semigroup – in Morrey spaces XXIV – in Morrey spaces with two indices XXIV – Schauder 182 estimate for blowup time 135

204 | Index

estimate for existence time 156 estimate of blowup time 92, 155 eternal solution 67 Feller test XVI fixed point 2 fractional Laplacian XIX, XXI – of power function 8 – of radially symmetric function 33 – of the indicator function of ball 44, 46 – scaling property 112 fractional power of Laplacian XVIII, XXI, 6, 30, 145, 166 free energy XIII Fujita method 6, 73, 137, 138 Fujita–Kato iterations 2 Gagliardo–Nirenberg inequality 29, 87, 181 Gauss–Stokes theorem 27, 46 Gauss–Weierstrass kernel XXIII, 65, 138 Hardy inequality 28, 178–180 Hardy–Littlewood maximal function 10 heat equation on the half line 26 Helmholtz equation 116, 119 Hessian of weight 80, 82, 111, 120 Hölder inequality in Morrey spaces 14 hypercontractivity 177, 181 ill-posedness of Cauchy problem 35, 155 implosion wave 74, 185 initial data – admissible XV, XVII, XIX, 2, 6, 7, 9, 86, 154, 157 initial trace of solution 154 integral kernel of semigroup XXII, XXIII integrated density XV interacting particle system 19, 77 interactions – attractive XI, 166 – attractive-repulsive XIII – electric XII – repulsive 166 Keller–Segel model – classical XI, 1, 86, 137, 157 – doubly parabolic XVI, XVIII, 58, 59, 160, 188 – minimal XI, 21 – parabolic-elliptic as limit XVII, 59 – with chemoattractant consumption XVII, 58, 115, 117, 118, 123, 160

– with fractional diffusion 32, 75, 117 – with nonlocal diffusion XI, XIX, 6, 30, 115, 131, 153 – with reaction terms XII kernel – mildly singular 166 – strongly singular 120 Lebesgue space XVIII Leray–Schauder method 188 Lévy diffusion operator XIX, 77, 166 Lévy–Khintchine representation XXII, 80, 81, 112, 147, 169 linearization operator 177 Lipschitz continuity of solution map 157, 158 logistic kinetics XIII Lyapunov functional XIV, 67 method of characteristics 167 mild solution XIV, 1, 19, 56, 59, 116, 124, 156, 159 moment of function – localized 163 – with Gaussian weight 138 moment of solution 162 – local 73, 87, 121 – rescaled 105, 112, 133 – second 76, 80, 87, 92 – truncated 120, 132 – with compactly supported weight function 92, 153 – with power weight 73, 75, 83, 170 monotone approximation 19 Moser–Trudinger inequality XIII multicomponent system XIII N-symmetric solution 88, 103 Navier–Stokes system XIII, XIV, 188 Nernst–Planck–Debye–Hückel system 71 Newtonian potential XI, 115, 120, 166 nonlinear diffusion XII, XIII, 19 nonlinear heat equation XIV, 138, 141 – with fractional diffusion XIV, 138, 146 nonlocal diffusion XIX, XXI nonlocal diffusion-transport equation XI, 6, 145 nonuniform diffusion XVI nonuniqueness of solutions 32, 60 norm continuous functions 3

Index | 205

parabolic regularization property 1, 3, 6, 10, 31, 50, 117 particle dynamics XI, 19 Picard iterations 4, 11, 12, 14, 15, 61 Poisson equation 16, 27 Poisson kernel of the ball 40 propagation of chaos property 19, 77 property ℐε 94, 96 quantization of mass XV, 31, 87 quenching 186 radial distribution function XV, 23 reflection method 26 regularization property – instantaneous XV, 155, 156 Rellich inequality 180 renormalization technique 19 Riesz potential 7 scaling property 7, 30, 76, 85, 89, 92, 102, 118, 158, 165 selfsimilar solution 29, 60, 68, 188 – blowing up 71 – doubly parabolic case XVII – nonuniqueness XVII, 60 – radial concentration 29 semigroup α/2

– e−t(−Δ) XXII – etΔ XXI sensitivity function XII, XIII separatrix 67 singular Gronwall lemma 5, 58 singular stationary solution 7 smoothness of solutions 6 Sobolev inequality 13, 120 solution oscillating between two steady states 188 solution with infinite mass 1

space – Besov XVIII −2 – homogeneous B∞,∞ (ℝd ) 144

−α – homogeneous B∞,∞ (ℝd ) 146 – Fourier–Besov–Morrey XVIII – Fourier–Herz XVIII – Lebesgue Lp (ℝd ) XX – Lorentz Lr,1 (ℝd ) 18 – Marcinkiewicz d d d Lw/2 (ℝd ) = L /2,∞ (ℝd ) = L /2,∗ (ℝd ) XVIII – Marcinkiewicz–Lorentz 18 – Marcinkiewicz–Morrey – homogeneous Msq,∗ (ℝd ) XX, 2, 10 – Morrey 13 – homogeneous Msq (ℝd ) XX – local 154 – of pseudomeasures 𝒫ℳa (ℝd ) XVIII – of Radon measures ℳ(ℝ2 ) XVIII, XX, 58, 60 – Sobolev W s,p (ℝd ), Hs = W s,2 XX spread of mass 91, 97 steady state XVI, XVIII – oscillation of density XVI, 68 Stirling formula 144, 150 Stroock–Varopoulos inequality 51 subcritical XIV, 31, 34, 43, 56 subsolution 181, 184, 187 successive approximations 2 sufficient condition for blowup 74, 108, 109, 117, 137 supercritical 31, 33, 36, 37 supercritical case XIV supersolution 181, 183 symmetrization technique 19

weak convergence XX, 3, 64, 65, 159 weak solution XIV, 1, 19, 77 weakly continuous functions 3 Weber–Fechner law XIII

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