Nonlinear parabolic equations and hyperbolic-parabolic coupled systems

This monograph is devoted to the global existence, uniqueness and asymptotic behaviour of smooth solutions to both initial value problems and initial boundary value problems for nonlinear parabolic equations and hyperbolic parabolic coupled systems. Most of the material is based on recent research carried out by the author and his collaborators.
The book can be divided into two parts. In the first part, the results on decay of solutions to nonlinear parabolic equations and hyperbolic parabolic coupled systems are obtained, and a chapter is devoted to the global existence of small smooth solutions to fully nonlinear parabolic equations and quasilinear hyperbolic parabolic coupled systems. Applications of the results to nonlinear thermoelasticity and fluid dynamics are also shown.
Some nonlinear parabolic equations and coupled systems arising from the study of phase transitions are investigated in the second part of the book. The global existence, uniqueness and asymptotic behaviour of smooth solutions with arbitrary initial data are obtained. The final chapter is further devoted to related topics: multiplicity of equilibria and the existence of a global attractor, inertial manifold and inertial set.
A knowledge of partial differential equations and Sobolev spaces is assumed. As an aid to the reader, the related concepts and results are collected and the relevant references given in the first
chapter. The work will be of interest to researchers and graduate students in pure and applied mathematics, mathematical physics and applied sciences.

Preface -- 1. Preliminaries -- 1.1. Basic Facts on Sobolev -- 1.1.1. Sobolev Spaces -- 1.1.2. The Gagliardo—Nirenberg and Poincaré Inequalities -- 1.1.3. Abstract Functions Valued in Banach Spacec -- 1.2. Linear Elliptic Equations -- 1.2.1. Boundary Value Problems -- 1.2.2. Interpolationpaces -- 1.3. Linear Evolution Equations -- 1.3.1. Variational Evolution Equations -- 1.3.2. Setting of Semigroup Theory -- 1.3,3. Hyperbolic and Parabolic Systems -- 1.4. Nonlinear Evolution Equations -- 1.4.1. Local Existence and Uniqueness -- 1.4.2. Global Existence -- 1.5. Asymptotic Behaviour -- 1.5.1. Global Attractor -- 1.5.2. Convergence to Equilibrium -- 1.5.3. Inertial Manifolds and Inertial Sets -- 1.6. Notation and Some Useful Inequalities -- 2. Decay of Solutions to Linear Evolution Equations -- 2.1. Linear Parabolic Equation -- 2.1.1. Initial Value Problems -- 2.1.2. Initial Boundar Value Problems -- 2.2. Linear Thermodastic Systems -- 2.2.1. Initial Value Problems -- 2.2.2. Initial Boundary Value Problems -- 2.3. Linear Thermoviscoelastic Systems -- 2.3.1. Initial Value Problems -- 2.3.2. Initial Boundary Value Problems -- 3. Global Existence for Small Initial Data . . . -- 3.1. Fully Nonlinear Parabolic Equations: IVP -- 3.2. Fully Nonlinear Parabolic E