We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L ∞ bounds for the weak solutions.
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Printed on acid-free paperThis work is dedicated to our families, Tamara, Nikolay, Stanislav and Elena, Dmitry, Andrey, to whom we owe so much.
PrefaceThis monograph is a contribution to the theory of second order quasilinear parabolic and hyperbolic equations with the nonlinear structure that may change from one point to another in the problem domain. In the past decade, there was an impetuous growth of interest in the study of such equations, which appear in a natural way in the mathematical modeling of various real-world phenomena and give rise to challenging mathematical problems. The aim of this work is to give an account of the known results on existence, uniqueness, and qualitative properties of solutions.The parabolic equations studied below can be conventionally divided into several groups. Chaps. 2 and 3 are devoted to study the generalized porous medium equationwith a given exponent mðx; tÞ [ À 1 and its generalizations, such as equations with lower order terms or anisotropic equations. We establish conditions of existence and uniqueness of weak solutions and show that for definite ranges of the exponent mðx; tÞ the solutions exhibit properties typical for the solutions of equations with constant m, those of the finite speed of propagation and extinction in finite time.The former means the following: if the support of the initial data is compact, then the support of the solution remains compact for all time but may expand in space with finite speed. The latter property means that the solution corresponding to a nonzero initial datum may extinct in a finite time. Chapters 4-6 concern the homogeneous Dirichlet problem for the nonlinear degenerate parabolic equations u t À divAðx; t; u; ruÞ þ Bðx; t; uÞ ¼ 0vii with the function A ðA 1 ; . . .; A n Þ whose components are of the formwhere p i ðx; tÞ 2 ð1; 1Þ are given measurable functions and the coefficients a i are Carathéodory functions, a i 2 ½a 0 ; a 1 with positive constants a 0 ; a 1 . The function B is assumed to satisfy the growth condition jBðx; t; rÞj djrj λ þ f ðx; tÞ with constants d ! 0, λ [ 1 and a given function f . Special attention is paid to the model case whenwith a continuous exponent σðx; tÞ 2 ð1; 1Þ and a given coefficient cðx; tÞ. Most of the results remain true if the operators A i are substituted by the Leray-Lions operators with variable...
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