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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Function Spaces

Abstract
This chapter is devoted to a brief exposition of the theory of function spaces that provide the analytic framework for the study of PDEs with variable nonlinearity.
Stanislav Antontsev, Sergey Shmarev

Chapter 2. A Porous Medium Equation with Variable Nonlinearity

Abstract
We devote this chapter to study the homogeneous Dirichlet problem for the semilinear parabolic equation which generalizes the Porous Medium Equation.
Stanislav Antontsev, Sergey Shmarev

Chapter 3. Localization of Solutions of the Generalized Porous Medium Equation

Abstract
Solutions of nonlinear parabolic equations may possess the properties not displayed by the solutions of any linear equation: the finite speed of propagation of disturbances from the problem data in space and vanishing in a finite time. In this chapter, we study the localization properties of solutions to gegenerate quasilinear parabolic PDEs with variable nonlinearity.
Stanislav Antontsev, Sergey Shmarev

Chapter 4. Anisotropic Equations with Variable Growth and Coercivity Conditions

Abstract
This chapter is devoted to study the Dirichlet problem for the evolution p(x,t)-Laplace equation and its generalizations. We derive sufficient conditions for existence and uniqueness of weak solutions.
Stanislav Antontsev, Sergey Shmarev

Chapter 5. Space Localization of Energy Solutions

Abstract
This chapter is devoted to study the properties of finite speed of propagation and space localization of solutions of parabolic PDEs with nonstandard growth.
Stanislav Antontsev, Sergey Shmarev

Chapter 6. Extinction in a Finite Time and the Large Time Behavior

Abstract
This chapter continues the study of the propagation properties of solutions of the Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity. We study the large time behavior and the possibility of extinction of solutions in a finite time.
Stanislav Antontsev, Sergey Shmarev

Chapter 7. Blow-up in Equations with Variable Nonlinearity

Abstract
A remarkable property of nonlinear evolution equations is the possibility of formation of singularities in solutions of problems with smooth input data, which leads to blow-up, or explosion, of initially regular solutions.
Stanislav Antontsev, Sergey Shmarev

Chapter 8. Equations with Double Isotropic Nonlinearity

Abstract
In this chapter we study the Dirichlet problem for the class of equations with double variable nonlinearity
Stanislav Antontsev, Sergey Shmarev

Chapter 9. Strong Solutions of Doubly Nonlinear Anisotropic Equations

Abstract
This chapter continues the study of doubly nonlinear equations.
Stanislav Antontsev, Sergey Shmarev

Chapter 10. Anisotropic Equations with Double Nonlinearity: Blow-up and Vanishing

Abstract
This chapter is devoted to the study of qualitative properties of strong solutions of the Dirichlet problem for the doubly nonlinear anisotropic parabolic equation with variable nonlinearity
Stanislav Antontsev, Sergey Shmarev

Chapter 11. Wave Equation with $$p(x,t)$$ p ( x , t ) -Laplacian

Abstract
In this chapter we study the homogeneous Dirichlet problem for the wave equation involving \(p(x,t)\)-Laplacian.
Stanislav Antontsev, Sergey Shmarev

Chapter 12. Semilinear Hyperbolic Equations

Abstract
In this chapter we study the Dirichlet problem for the semilinear hyperbolic equation
Stanislav Antontsev, Sergey Shmarev

Backmatter

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