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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

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For the past several decades, the study of free boundary problems has been a very active subject of research occurring in a variety of applied sciences. What these problems have in common is their formulation in terms of suitably posed initial and boundary value problems for nonlinear partial differential equations. Such problems arise, for example, in the mathematical treatment of the processes of heat conduction, filtration through porous media, flows of non-Newtonian fluids, boundary layers, chemical reactions, semiconductors, and so on. The growing interest in these problems is reflected by the series of meetings held under the title "Free Boundary Problems: Theory and Applications" (Ox­ ford 1974, Pavia 1979, Durham 1978, Montecatini 1981, Maubuisson 1984, Irsee 1987, Montreal 1990, Toledo 1993, Zakopane 1995, Crete 1997, Chiba 1999). From the proceedings of these meetings, we can learn about the different kinds of mathematical areas that fall within the scope of free boundary problems. It is worth mentioning that the European Science Foundation supported a vast research project on free boundary problems from 1993 until 1999. The recent creation of the specialized journal Interfaces and Free Boundaries: Modeling, Analysis and Computation gives us an idea of the vitality of the subject and its present state of development. This book is a result of collaboration among the authors over the last 15 years.

Boundary value problems. --- Differential equations, Partial. --- Fluid mechanics --- Mathematics. --- Boundary value problems --- Differential equations, Partial --- Mathematics --- Fluid mechanics. --- Partial differential equations. --- Functional analysis. --- Applied mathematics. --- Engineering mathematics. --- Mechanics. --- Engineering Fluid Dynamics. --- Partial Differential Equations. --- Functional Analysis. --- Applications of Mathematics. --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Engineering --- Engineering analysis --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Hydromechanics --- Continuum mechanics --- Fluid mechanics - Mathematics