Proceedings of the Fourth International Colloquium on Differential Equations, Plovdiv, Bulgaria, 18–22 August 1993 [Reprint 2020 ed.] 9783112318874, 9783112307601

Citation preview

Proceedings of the Fourth International Colloquium on Differential Equations

Proceedings of the Fourth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 2 August 1993

Editors: D. Bainov and V. Covachev

///VSP///

Utrecht, The Netherlands, 1994

VSP BV P.O. B o x 3 4 6 3 7 0 0 A H Zeist The Netherlands

© V S P B V 1994 First p u b l i s h e d in 1 9 9 4 ISBN 90-6764-169-3

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

by Koninklijke

Wöhrmann,

Zutphen

CONTENTS Preface

1

Periodic Solutions for Multi-Phase Stefan Problems with Dynamic Boundary Conditions T. Aiki

3

Overdetermined Problems for Elliptic Equations D. Alemayehu and G. Porru

11

Mixed Finite-Element Methods for the Equations of Flow in Porous Media M.B. Allen

21

Limit Sets of Impulsive Dynamical Systems D.D. Bainov and S.I. Nenov

31

Non-Classical Dynamic Programming in Optimal Control S.A. Belbas

39

Painlevé Analysis of Nonlinear Reaction-Diffusion Guo Ben-yu

45

Weighted Interpolation and Hardy Inequalities with Some Spectral Theoretic Applications R. C. Brown and D.B. Hinton

55

Relativity and Gravitation: Probabilistic Germs and Survey J. P. Caubet

71

Monodromic Unbounded Polycycles M. Galeotti

83

Uniform Schemes for Singularly Perturbed Scalar Initial Value Problems P. González-Vera, S. González-Pinto, andL. Casasús

95

The Operator Theoretical Treatment of Partial Differential Equations: Recent Developments with Applications M. Grobbelaar

105

Hilbert-Arnold Problem for Cubic Hamiltonians and Limit Cycles E. Horozov and I.D. Iliev

115

Periodic Solutions of Neutral Impulsive Systems with a Small Delay L. Jodar, R.J. Villanueva, and V.C. Covachev

125

vi

contents

Analytic-Numerical Solutions of Variable Coefficient Second Order Delay Differential Equations L. Jodar, R.J. Villanueva, V.C. Covachev, and J. A. Martin

137

Localization of Global Existence of Holomorphic Solutions of Differential Equations with Complex Parameters J. Kajiwara

147

Conditions for the Absence of Positive Solutions of First Order Differential Inequalities with Deviating Arguments E. Kozakiewicz

157

On the Solution of Boundary Value Problem. The Decomposition Method E.S. Lee and G. Adomian

163

Some Uniform Estimates for Solutions of Difference Differential Equations of Elliptic-Parabolic Type M. Misawa

167

The Eigenvalue Distribution of Elliptic Operators with Holder Continuous Coefficients Y. Miyazaki

175

Applications of an Adaptive Finite R. Montenegro, L. Ferragut, and A.Element Plaza Method

189

Formation of Shocks for Nonlinear Partial Differential Equations of First Order S. Nakane

199

Some Degenerate Nonlinear Parabolic Equations T. Nanbu

205

Existence Theory for Singular Two Point Boundary Value Problems D. O'Regan

215

Microlocal Hypoellipticity of Some Classes of Overdetermined Systems of Pseudodifferential Operators P.R. Popivanov

229

A Two Point Connection Problem for a Certain Third Order Differential Equation T.K. Puttaswamy and T. V. Sastry Strong-Electric-Field Eigenvalue Asymptotics for the Perturbed Magnetic Schrodinger Operator G.D. Raikov

235

245

contents

vii

Stability Theory for a Class of 2D Linear Systems Described by a Set of Recursive Differential Equations E. Rogers and D.H. Owens

253

Analyticity of the Solutions of the Heat Equation on the Half space RJ S. Saitoh

265

Recent Advances in the Method of Quasilinearization A.S. Vatsala

277

On a Radially Symmetric Minimal Surface with Non-Empty Free Boundary Y. Yamaura

287

Sampling Theorems Associated with Boundary-Value Problems A.I. Zayed

295

Preface T h e Fourth International Colloquium on Differential Equations was organized by U N E S C O and the Mathematical Society o f Japan, with the cooperation o f the Association Suisse d'lnformatique, the Canadian Mathematical Society, Hamburg University, Kyushu University, the London Mathematical Society, Plovdiv University "Paissii Hilendarski", Technical University - Berlin, Technical University Plovdiv, Union o f the Bulgarian Mathematicians -Plovdiv Section, the University o f T e x a s -Pan American, and sponsored by the firm "Eureka". It was held on August 18-23, 1993, in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 18-23 August in Plovdiv, Bulgaria. The address o f the organizing committee is: Stoyan Zlatev, Mathematical Faculty o f the Plovdiv University, Tsar Assen Str.24, Plovdiv 4 0 0 0 , Bulgaria.

The Editors

4th Int. Coll. on Differential Equations, pp. 3-9 D. Bainov and V. Covachev (Eds) © VSP 1994

Periodic solutions for multi-phase Stefan problems with dynamic boundary conditions TOYOHIKO AIKI Department of Mathematics Faculty of Education, Gifu University 1-1, Yanagido, Gifu, 501-11, Japan

A B S T R A C T . We consider periodic solutions to multi-phase Stefan problems for a class of nonlinear parabolic equations with dynamic boundary conditions, that is, with boundary conditions containing time derivatives on the fixed boundary. In this paper, we investigate existence, order property and asymptotic stability of periodic solutions. Key words: periodic solutions, dynamic boundary conditions, asymptotic behavior Introduction In this paper, we consider periodic stabilities for the solutions to Stefan problems with periodic condition in time in the enthalpy formulation with dynamic boundary conditions: Find functions u = u(t, x) on R x ft and V = V(t, x) on R x T satisfying that ut - Aß(u) = 0

in fi x ft,

(o.i) (0.2)

where ft is a bounded domain in R N(N > 2) with smooth boundary T = 9ft; (3 : R —* R is a given nondecreasing function; g = g(t,x,£) : R x T x R R is a given function which is nondecreasing in r € R for a.e. (2, x) € R x T; (d/dv) denotes the outward normal derivative on I\ We denote by SP — SP(/3,g) the above system (0.1) ~ (0.2). By many authors initial-boundary value problems for (0.1) with usual boundary conditions have been studied. In case the flux condition is of the form —d/3(u)/du = g(t,x,/3(u)), the problem was uniquely solved in the variational sense by Visintin [1], Niezgodka, Pawlow & Visintin [2] and Niezgodka & Pawlow [3]. Also, some interesting results dealing with the boundary condition (0.2) are found in Cannon [4], Hintermann [5] and Grobbelaar Dalsen [6]. Recently, boundary conditions similar to (0.2) were discussed by Primicerio & Rodrigues [7] and in Aiki [8] for one-dimensional Stefan problems with dynamic boundary conditions the local in time existence and uniqueness of classical solutions were shown. In particular, the existence and uniqueness for the

T. Aiki

4

SP(/3,g)

solution to the initial-boundary value problems for axe studied in Aiki [9]. The purpose of this paper is to establish the existence, the order property and the asymptotic stability of periodic solutions under the periodic condition in time, that is, a.e. on for some positive constant In papers by Haraux and Kenmochi [10], Aiki, Kenmochi and Shinoda [11, 12] and Aiki [13] similar questions to those above were discussed. In this paper, we omit proofs of main results and shall show those in the author's forthcoming paper [14].

g(t,x,£) = g(t + T,x,£)

RxTxR

T.

1. S t a t e m e n t s of main results Throughout this paper we assume that the function /? : R —• R satisfies the following conditions (/?1) and (/?2): (/Si) ft is non-decreasing and Lipschitz continuous on R with /?(0) = 0. (/?2) There are some positive constants Lp, lp such that I0(r)| > b\r\ ~ b

^ r all r € R.

g = g(t,x,£) : R xT x R—t R g(t,x,£) ( R, € LjJRg(t,x,()

Also, let be a function and suppose that: (gl) is non-decreasing in £ £ -ft for a.e. (t,x) G xT; (g2) for any e g(;-,0 L2(r)); (g3) is locally Lipschitz continuous in £ uniformly with respect to R x T, that is, for each M > 0 there is a positive constant Cg(M) such that

for all with |£| < M, with mi < m 2 such that

< M and a.e. ( i , x ) € RxT

R

(t,x) €

and there are constants m i , m 2 (t,x) 6 E.

g(t,x,fl(mi))< 0,g(t,x,/S(m2)) > 0 for a.e.

For the sake of simplicity of notations we put

X = Hl{il), H = i2(H) x L2{T),

A(u,v)J= n / Vu • VWx for u,v £ X, (u,v)x = A(u,v) + (J udx + j udT)(J vdx + J vdT) for u,v € X, (u, v)n = I uv dx + I upvp H by putting

Ev =

(v, u|r)

for

v € X.

Clearly, X and H are Hilbert spaces with inner product (•, and (•, -)H, respectively, and the range of E, R(E), is a dense subspace of H and E is linear and compact. We

Stefan problems

with dynamic

B.

C.

identify H with its dual H* and denote by X* the dual space of X. Therefore, denoting by E' the dual operator of E we have (E"(v,vr),r))x

= I vt] dx + I vpr) dT Jn

Jr

for any ( v , f r ) £ H and tj £ X,

is a duality pairing between X* and X.

where (•,

We now formulate problem S P in variational sense. Definition 1.1. Let J = [ a0 > I u2(0,xWx+ / V2(0,xWr

Ja

Jr

Jn

Jr

there exists a T-periodic solution {u, V} of SP on R such that a0=

[ u(0,x)dx Jn

+ / Jr

V(0,x) /?(«) € L2(0, T; Wl'2(Sl)) as n ^ oo.

2. Known results In this section we show some known results which will use in the proof of theorems 1.2 ~ 1.5 First, we recall a comparison result for CSP(u0, V0). Theorem 2.1. (cf. [9, Theorem 1.3]) Suppose that the same assumptions as in Theorem 1.1 hold and a pair {u0, Vo} € L°°(Sl) x £°°(r). Let t0 be any number in R and {u,V} (resp. {u, V}) be a solution of CSP(uo,Vo) (resp. CSP(uo,Vo)) on[t 0 ,oo). Then for any t € [i0, oo), Mt)-m+\om

\W(t)-v(t)]+\LllD

|[u(io) - u(to)}+\LHil) + \{V(t0) - V(t0))+\LHn.

< In particular, ifui(t0,-) ui < u 2

+

< u2(t0,-) a.e. on SI andVi(tQ,-) a.e. on R x SI

and

Vj < V2

(2.1)

< (i 0 , •) a.e. on T, then o.e. on R x I\

Next, we consider the degenerate parabolic equation with linear dynamic boundary condition: ut — Af){u) = 0 in (t 0 , oo) x SI, ' ^

+

% + *= »

/3(u) = V 2

on (i 0 , oo) x T,

(2-2)

on (t0, oo) x T, 2

where h is a function given in L 0C(R; £ ( r ) ) and —oo < i 0 < oo; : R —• ii is a function satisfying (>91) and (/32). Following [13, section2], we say that for any compact interval J = [¿ 0 ,i^, a couple of functions u : J L2(Sl) and V : J —> L2(T)) is a solution of (2.2) on J, if E*(u, V) € ^ ^ ( J ; ^ ) , u € L°°(J-,L2(Sl)), P(u) e L2(J\X), V € L°°(J; L2(T)), /3(u) = V a.e. on J x T and (^Em(u(t),V(t)),t])x+A(P(u(t)),T])+j^h(t,-)iidT

= 0

for any 77 € X and a.e. t € J.

For a general interval J' C R, solutions of (2.2) on J' are defined in a manner similar to definition 1.2. Also, solutions of Cauchy problems and the problems with T-periodic condition are defined just as definition 1.3. Next, we mention some results on T-periodic solutions to (2.2) on R. Theorem 2.2. We suppose that h £ Lfoc(R; L2(T)). Let T be a positive number, and assume that h(t + T, •) = h(t, •) a.e. on T for any t