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Über dieses Buch

This book presents a number of analytic inequalities and their applications in partial differential equations. These include integral inequalities, differential inequalities and difference inequalities, which play a crucial role in establishing (uniform) bounds, global existence, large-time behavior, decay rates and blow-up of solutions to various classes of evolutionary differential equations. Summarizing results from a vast number of literature sources such as published papers, preprints and books, it categorizes inequalities in terms of their different properties.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Integral Inequalities

It is well known that classical integral inequalities furnishing explicit bounds for an unknown function play a fundamental role in differential and integral equations. In this chapter, we shall first collect some basic integral inequalities.
Yuming Qin

Chapter 2. Differential and Difference Inequalities

In this chapter, we establish differential and difference inequalities in analysis that play a role in applications in the subsequent chapters.
Yuming Qin

Chapter 3. Attractors for Evolutionary Differential Equations

In this chapter, we prove the existence of global (uniform) attractors for some evolutionary differential equations. The chapter includes four sections. In Section 3.1, we shall use Theorems 1.1.2 and 2.1.3 to establish the existence of global attractors for a nonlinear reaction-diffusion equation.
Yuming Qin

Chapter 4. Global Existence and Uniqueness for Evolutionary PDEs

In this chapter, we present some results on global existence and uniqueness of solutions to evolutionary PEDs obtained by application of analytic inequalities in Chapters 1 and 2. This chapter consists of four sections. In Section 4.1, we use the simultaneous singular Bellman–Gronwall inequality, i.e., Theorem 1.3.2, to discuss the local existence, regularity, and continuous dependence on initial data of solutions to a weakly coupled parabolic system for non-regular initial data. In Section 4.2, we use Theorem 1.4.9 to study some properties of solutions to the Cauchy problem for multi-dimensional conservation laws with anomalous diffusion. In Section 4.3, we use Theorem 2.1.19 to investigate the blow-up of solutions of semilinear heat equations.
Yuming Qin

Chapter 5. Global Existence and Uniqueness for Abstract Evolutionary Differential Equations

In this chapter, we discuss the global existence and uniqueness for some abstract models and ODEs. The chapter consists of three sections. In Section 5.1, we apply Corollary 1.4.4 to prove the global-in-time existence of solutions to an abstract evolutionary equation written below.
Yuming Qin

Chapter 6. Global Existence and Asymptotic Behavior for Equations of Fluid Dynamics

In this chapter, we prove the global existence and asymptotic behavior of solutions to fluid models. The chapter includes five sections. In Section 6.1, we exploit Theorems 1.4.11 and 1.5.1 to show the asymptotic behavior of the solutions to the Navier–Stokes equations in 2D exterior domains.
Yuming Qin

Chapter 7. Asymptotic Behavior of Solutions for Parabolic and Elliptic Equations

In this chapter, we shall study the asymptotic behavior for parabolic and elliptic equations. This chapter embraces three sections. In Section 7.1, we shall use Theorems 2.1.14 and 2.3.7 to establish the uniform and decay estimates for flows in a semi-infinite straight channel. In Section 7.2, we shall exploit Theorems 2.3.17–2.3.21 to establish exact rates of convergence for nonlinear PDEs.
Yuming Qin

Chapter 8. Asymptotic Behavior of Solutions to Hyperbolic Equations

This chapter mainly studies the asymptotic behavior of solutions to some hyperbolic equations. This chapter includes seven sections. In Section 8.1, we shall use Theorem 2.3.11 to study the decay of solutions to 1D nonlinear wave equations. In Section 8.2, we shall exploit Theorem 2.3.14 to investigate the decay property of the solutions to the initial boundary value problem for a wave equation with a dissipative term. In Section 8.3, we shall apply Theorem 2.3.6 to establish the polynomial decay rate for nonlinear wave equations.
Yuming Qin

Chapter 9. Asymptotic Behavior of Solutions to Thermoviscoelastic, Thermoviscoelastoplastic and Thermomagnetoelastic Equations

In this chapter, we shall establish the asymptotic behavior for thermoviscoelastic, thermoviscoelastoplastic and thermomagnetoelastic equations. This chapter consists of three sections. In Section 9.1, we shall first employ Lemma 1.5.4 to extend the decay results in [620] for a viscoelastic system to those for the thermoviscoelastic system (9.1.1) and then establish the existence of the global attractor for the homogeneous thermoviscoelastic system (9.1.54).
Yuming Qin

Chapter 10. Blow-up of Solutions to Nonlinear Hyperbolic Equations and Hyperbolic-Elliptic Inequalities

In this chapter, we shall consider the blow-up of solutions to some nonlinear hyperbolic equations and hyperbolic-elliptic inequalities. This chapter consists of seven sections. In Section 10.1, we apply Theorem 2.4.6 to investigate the blow-up of solutions to semilinear wave equations. In Section 10.2, we shall employ Theorem 2.4.22 to study the blow-up of solutions to semilinear wave equations.
Yuming Qin

Chapter 11. Blow-up of Solutions to Abstract Equations and Thermoelastic Equations

In this chapter, we shall study the blow-up of solutions to abstract equations and thermoelastic equations. This chapter consists of six sections. In Section 11.1, we shall employ Theorem 2.4.19 to prove the blow-up results of solutions to a class of abstract initial and initial boundary value problems. In Section 11.2, we shall employ Theorem 2.4.20 to study the blow-up of solutions to a class abstract nonlinear equations.
Yuming Qin

Chapter 12. Appendix: Basic Inequalities

In this chapter, we shall collect some basic inequalities which play a very crucial role in classical calculus. These inequalities include the Young inequality, the Hölder inequality, the Minkowski inequality, the Jensen inequality, and the Hausdorff–Young inequality, etc.
Yuming Qin

Backmatter

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