Evolution PDEs with Nonstandard Growth Conditions Existence, Uniqueness, Localization, Blow-up

This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.

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 ?nite 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 p(x, t )-Laplacian.- Semilinear hyperbolic equations.