Analysis of Reaction-Diffusion Models with the Taxis Mechanism

Preface
Contents
1 Chemotaxis–Fluid System
1.1 Introduction
1.2 Preliminaries
1.3 Global Boundedness of Solution to a Chemotaxis …
1.3.1 A Quasi-energy Functional
1.3.2 upper L Superscript normal infinity Baseline left parenthesis left parenthesis 0 comma normal infinity right parenthesis semicolon upper L Superscript p Baseline left parenthesis upper Omega right parenthesis right parenthesisLinfty((0,infty);Lp()) Estimate of n Subscript epsilonnε for Some p greater than three halvesp > 32
1.3.3 Uniform upper L Superscript normal infinityLinfty-Boundedness of n Subscript epsilonnε as Well as nabla c Subscript epsiloncε and u Subscript epsilonuε
1.3.4 Global Boundedness of Weak Solutions
1.4 Asymptotic Profile of Solution to a Chemotaxis …
1.4.1 Basic a Priori Bounds
1.4.2 Global Boundedness of Solutions
1.4.3 Asymptotic Profile of Solutions
2 Keller–Segel–Navier–Stokes System Involving Tensor-Valued Sensitivity
2.1 Introduction
2.2 Preliminaries
2.3 Blow-Up Prevention by Nonlinear Diffusion …
2.3.1 Some Basic a Priori Estimates
2.3.2 Global Boundedness of Weak Solutions
2.4 Global Existence of Solutions to a Three-Dimensional Keller …
2.4.1 A Priori Estimates for Approximate Solutions
2.4.2 Global Solvability of the Approximate System
2.4.3 Regularity Property of Time Derivatives
2.4.4 Global Existence of Weak Solutions
3 Chemotaxis–Haptotaxis System
3.1 Introduction
3.2 Preliminaries
3.3 Global Boundedness of Solutions to a Chemotaxis–Haptotaxis Model
3.4 Global Boundedness of Solutions to a Chemotaxis …
3.4.1 A Convenient Extensibility Criterion
3.4.2 Global Existence in Two-Dimensional Domains
3.4.3 Global Existence in Three-Dimensional Domains
3.5 Asymptotic Behavior of Solutions to a Chemotaxis–Haptotaxis Model
3.5.1 Global Boundedness
3.5.2 Asymptotic Behavior
4 Keller–Segel–(Navier–)Stokes System Modeling Coral Fertilization
4.1 Introduction
4.2 Preliminaries
4.3 Global Boundedness and Decay Property of Solutions to a 3D …
4.3.1 A Convenient Extensibility Criterion
4.3.2 Global Boundedness and Decay for script upper S equals 0mathcalS=0 on partial differential upper Omega
4.3.3 Global Boundedness and Decay for General script upper SmathcalS
4.4 Asymptotic Behavior of Solutions to a Coral Fertilization Model
4.4.1 A Convenient Extensibility Criterion
4.4.2 Global Boundedness and Decay for script upper S equals 0mathcalS=0 on partial differential upper Omega
4.4.3 Global Boundedness and Decay for General script upper SmathcalS
4.5 Large Time Behavior of Solutions to a Coral Fertilization …
4.5.1 Regularized Problems
4.5.2 Conditional Uniform Bounds for left parenthesis nabla c Subscript epsilon Baseline right parenthesis Subscript epsilon element of left parenthesis 0 comma 1 right parenthesis(cε)εin(0,1)
4.5.3 A Prior Estimates
4.5.4 Global Solvability
4.5.5 Asymptotic Behavior
5 Density-Suppressed Motility System
5.1 Introduction
5.2 Preliminaries
5.3 Traveling Wave Solutions to a Density-Suppressed Motility Model
5.3.1 Some a Priori Estimates
5.3.2 Auxiliary Problems
5.3.3 Minimal Wave Speed
5.3.4 Selection of Wave Profiles
5.4 Asymptotic Behavior of Solutions to a Signal-Suppressed Motility Model
5.4.1 Space–Time upper L Superscript 1L1-Estimates for u Subscript epsilon Superscript m plus 1 Baseline v Subscript epsilon Superscript negative alphaum+1εv-αε
5.4.2 Boundedness of Solutions left parenthesis u Subscript epsilon Baseline comma v Subscript epsilon Baseline comma w Subscript epsilon Baseline right parenthesis(uε, vε, wε)
5.4.3 Asymptotic Behavior
6 Multi-taxis Cross-Diffusion System
6.1 Introduction
6.2 Preliminaries
6.3 Asymptotic Behavior of Solutions to a Doubly Tactic Resource Consumption Model
6.4 Boundedness of Solutions to an Oncolytic Virotherapy Model
6.4.1 Some Basic a Prior Estimates
6.4.2 Bounds for aa, bb and cc in upper L l o g upper L Llog L
6.4.3 upper L Superscript normal infinityLinfty-Bounds for aa, bb and cc
6.5 Asymptotic Behavior of Solutions to an Oncolytic Virotherapy Model
Appendix References