Analysis of Reaction-Diffusion Models with the Taxis Mechanism

Description

This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a  priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics. 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