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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.

Mathematics. --- Partial Differential Equations. --- Functional Analysis. --- Functional analysis. --- Differential equations, partial. --- Mathématiques --- Analyse fonctionnelle --- Differential equations, Hyperbolic. --- Differential equations, Parabolic. --- Quasilinearization. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Parabolic differential equations --- Parabolic partial differential equations --- Hyperbolic differential equations --- Partial differential equations. --- Differential equations, Nonlinear --- Differential equations, Partial --- Numerical solutions --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations