## Analytic Inequalities and Their Applications in PDEsThis 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. |

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### Contents

1 | |

Chapter 2 Differential and Difference Inequalities | 79 |

Chapter 3 Attractors for Evolutionary Differential Equations | 209 |

Chapter 4 Global Existence and Uniqueness for Evolutionary PDEs | 231 |

Chapter 5 Global Existence and Uniqueness for Abstract Evolutionary Differential Equations | 245 |

Chapter 6 Global Existence and Asymptotic Behavior for Equations of Fluid Dynamics | 257 |

Chapter 7 Asymptotic Behavior of Solutions for Parabolic and Elliptic Equations | 291 |

Chapter 8 Asymptotic Behavior of Solutions to Hyperbolic Equations | 307 |

Chapter 9 Asymptotic Behavior of Solutions to Thermoviscoelastic Thermoviscoelastoplastic and Thermomagnetoelastic Equations | 357 |

Chapter 10 Blowup of Solutions to Nonlinear Hyperbolic Equations and HyperbolicElliptic Inequalities | 393 |

Chapter 11 Blowup of Solutions to Abstract Equations and Thermoelastic Equations | 429 |

Basic Inequalities | 466 |

Bibliography | 497 |

561 | |

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a e Q absorbing set Appl apply Theorem Assume assumptions asymptotic behavior attractors Banach space Bellman-Gronwall inequality blow-up of solutions boundary conditions bounded Cauchy problem completes the proof continuous function convex convex function Corollary decay rate defined denote derive differential inequality dissipative domain eaſists energy estimate evolution equations finite following result Global existence global solutions H-co H.A. Levine heat equation Hence Hilbert space Hº Q Hölder inequality holds i-Co implies initial data integral inequalities Jensen inequality Lemma linear Math Nakao Navier–Stokes equations non-decreasing non-negative function Nonlinear Anal nonlinear wave equations O J RN obtain one-dimensional parabolic equations Partial Differential Equations positive constants quasilinear satisfies semigroup semilinear ſº solution of problem strictly increasing t–HT t—HT Theorem viscous wave equations yields