Nonlinear Second Order Parabolic Equations [1 ed.] 2020042644, 2020042645, 9780367711986, 9780367712846, 9781003150169

Table of contents :
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

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