Nonlinear Second Order Parabolic Equations

Description

The parabolic partial differential equations model one of the most important processes in the real-world: diffusion. Whether it is the diffusion of energy in space-time, the diffusion of species in ecology, the diffusion of chemicals in biochemical processes, or the diffusion of information in social networks, diffusion processes are ubiquitous and crucial in the physical and natural world as well as our everyday lives. This book is self-contained and covers key topics such as the Lp theory and Schauder theory, maximum principle, comparison principle, regularity and uniform estimates, initial-boundary value problems of semilinear parabolic scalar equations and weakly coupled parabolic systems, the upper and lower solutions method, monotone properties and long-time behaviours of solutions, convergence of solutions and stability of equilibrium solutions, global solutions and finite time blowup. It also touches on periodic boundary value problems, free boundary problems, and semigroup theory. The book covers major theories and methods of the field, including topics that are useful but hard to find elsewhere. This book is based on tried and tested teaching materials used at the Harbin Institute of Technology over the past ten years. Special care was taken to make the book suitable for classroom teaching as well as for self-study among graduate students. Cover Half Title Title Page Copyright Page Contents Preface Chapter 1: Preliminaries 1.1. NOTATIONS, AGREEMENTS AND BASIC ASSUMPTIONS 1.2. THE Lp AND SCHAUDER THEORIES 1.2.1. Lp theory 1.2.2. Schauder theory 1.2.3. Existence and uniqueness of solutions of linear system 1.3. MAXIMUM PRINCIPLE OF CLASSICAL SOLUTIONS 1.3.1. Initial-boundary value problems 1.3.2. Cauchy problems 1.3.3. Weakly coupled systems 1.3.4. Equations with non-classical boundary conditions 1.3.5. Equations with non-local terms 1.4. MAXIMUM PRINCIPLE OF STRONG SOLUTIONS 1.5. WEAK SOLUTION EXERCISES Chapter 2: Comparison Principle, Regularity and Uniform Estimates 2.1. COMPARISON PRINCIPLE 2.1.1. Classical and strong solutions 2.1.2. Weak solutions 2.1.3. Quasilinear equations with nonlinear boundary conditions 2.2. REGULARITY AND UNIFORM ESTIMATES 2.3. UNIFORM BOUNDS FROM BOUNDED Lp NORMS EXERCISES Chapter 3: Semilinear Parabolic Equations 3.1. THE UPPER AND LOWER SOLUTIONS METHOD 3.2. SOME EXAMPLES 3.3. MONOTONICITY AND CONVERGENCE 3.4. THE WEAK UPPER AND LOWER SOLUTIONS METHOD EXERCISES Chapter 4: Weakly Coupled Parabolic Systems 4.1. LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS 4.2. THE UPPER AND LOWER SOLUTIONS METHOD 4.2.1. Existence and uniqueness of solutions 4.2.2. Monotone iterative 4.3. APPLICATIONS 4.3.1. A competition model 4.3.2. A prey-predator model 4.3.3. The Belousov-Zhabotinskii reaction model EXERCISES Chapter 5: Stability Analysis 5.1. LOCAL STABILITY DETERMINED BY THE PRINCIPAL EIGENVALUE 5.2. LYAPUNOV FUNCTIONAL METHOD 5.2.1. Abstract results 5.2.2. A variable-territory prey-predator model 5.2.3. A prey-predator model with trophic interactions 5.2.4. A system coupled by two PDEs and one ODE 5.3. ITERATION METHOD 5.4. AVERAGE METHOD 5.4.1. Abstract results 5.4.2. A special diffusive competition model EXERCISES Chapter 6: Global Existence and Finite Time Blowup 6.1. SOME BASIC METHODS 6.1.1. Comparison method 6.1.2. Kaplan’s first eigenvalue method 6.1.3. Energy method 6.1.4. Concavity method 6.2. A SYSTEM WITH DIRICHLET BOUNDARY CONDITIONS 6.2.1. Critical exponents 6.2.2. Blowup rate estimates 6.3. A SYSTEM COUPLED IN EQUATION AND BOUNDARY 6.3.1. Properties of the Green function G(x, y, t − s) 6.3.2. Critical exponents 6.3.3. Blowup on the boundary 6.3.4. Blowup rate estimates EXERCISES Chapter 7: Time-Periodic Parabolic Boundary Value Problems 7.1. THE UPPER AND LOWER SOLUTIONS METHOD FOR SCALAR EQUATIONS 7.2. TIME-PERIODIC PARABOLIC EIGENVALUE PROBLEMS 7.3. THE LOGISTIC EQUATION 7.3.1. The case of bounded domain 7.3.2. The case of half line 7.4. THE UPPER AND LOWER SOLUTIONS METHOD FOR SYSTEMS 7.5. A DIFFUSIVE COMPETITION MODEL EXERCISES Chapter 8: Free Boundary Problems from Ecology 8.1. DEDUCTION OF FREE BOUNDARY CONDITIONS 8.2. EXISTENCE AND UNIQUENESS OF SOLUTIONS 8.3. REGULARITY AND UNIFORM ESTIMATES 8.4. SOME TECHNICAL LEMMAS 8.5. THE LOGISTIC EQUATION 8.5.1. Longtime behaviour of solution 8.5.2. Criteria and dichotomy for spreading and vanishing 8.6. SPREADING SPEED OF FREE BOUNDARY 8.7. A SYSTEM COUPLED BY ONE PDE AND ONE ODE EXERCISES Chapter 9: Semigroup Theory and Applications 9.1. C0 SEMIGROUP THEORY 9.1.1. Basic results 9.1.2. Solutions of linear problems 9.1.3. Mild solutions of semilinear problems 9.2. ANALYTIC SEMIGROUP THEORY 9.2.1. Basic results 9.2.2. Regularity of function Z t 0e−(t−s)Af(s)ds 9.2.3. Real solutions of semilinear problems 9.3. SEMIGROUPS DETERMINED BY − IN Lp( ) 9.4. AN EXAMPLE EXERCISES Appendix A: Appendix Bibliography Index