Antontsev S., Shmarev S. Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up

Atlantis Press, 2015. — 417 p. — (Atlantis Studies in Differential Equations 04). — ISBN 978-94-6239-111-6.This 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 Function Spaces
A Porous Medium Equation with Variable Nonlinearity
Localization of Solutions of the Generalized Porous Medium Equation
Anisotropic Equations with Variable Growth and Coercivity Conditions
Space Localization of Energy Solutions
Extinction in a Finite Time and the Large Time Behavior
Blow-up in Equations with Variable Nonlinearity
Equations with Double Isotropic Nonlinearity
Strong Solutions of Doubly Nonlinear Anisotropic Equations
Anisotropic Equations with Double Nonlinearity: Blow-up and Vanishing
Wave Equation with pðx; tÞ-Laplacian
Semilinear Hyperbolic Equations