Nonlinear Systems of Partial Differential Equations : Applications to Life and Physical Sciences.
Title:
Nonlinear Systems of Partial Differential Equations : Applications to Life and Physical Sciences.
Author:
Leung, Anthony W.
ISBN:
9789814277709
Personal Author:
Physical Description:
1 online resource (545 pages)
Contents:
Contents -- Preface -- 1 Positive Solutions for Systems of Two Equations -- 1.1 Introduction -- 1.2 Strictly Positive Coexistence for Diffusive Prey-Predator Systems -- 1.3 Strictly Positive Coexistence for Diffusive Competing Systems -- 1.4 Strictly Positive Coexistence for Diffusive Cooperating Systems -- 1.5 Stability of Steady-States as Time Changes -- Part A: Prey-Predator Case. -- Part B: Competing Species Case. -- 2 Positive Solutions for Large Systems of Equations -- 2.1 Introduction -- 2.2 Synthesizing Large (Biological) Diffusive Systems from Smaller Subsystems -- 2.3 Application to Epidemics of Many Interacting Infected Species -- 2.4 Conditions for Coexistence in Terms of Signs of Principal Eigenvalues of Related Single Equations, Mixed Boundary Data -- 2.5 Positive Steady-States for Large Systems by Index Method -- 2.6 Application to Reactor Dynamics with Temperature Feedback -- 3 Optimal Control for Nonlinear Systems of Partial Differential Equations -- 3.1 Introduction and Preliminary Results for Scalar Equations -- 3.2 Optimal Harvesting-Coefficient Control of Steady-State Prey- Predator Diffusive Volterra-Lotka Systems -- 3.3 Time-Periodic Optimal Control for Competing Parabolic Systems -- 3.4 Optimal Control of an Initial-Boundary Value Problem for Fission Reactor Systems -- 3.5 Optimal Boundary Control of a Parabolic Problem -- 4 Persistence, Upper and Lower Estimates, Blowup, Cross-Diffusion and Degeneracy -- 4.1 Persistence -- 4.2 Upper-Lower Estimates, Attractor Set, Blowup -- 4.3 Diffusion, Self and Cross-Diffusion with No-Flux Boundary Condition -- 4.4 Degenerate and Density-Dependent Diffusions, Non-Extinction in Highly Spatially Heterogenous Environments -- Part A: Weak Upper and Lower Solutions for Degenerate or Non- Degenerate Elliptic Systems.
Part B: Lower Bounds for Density-Dependent Di.usive Systems with Regionally Large Growth Rates. -- 5 TravelingWaves, Systems ofWaves, Invariant Manifolds, Fluids and Plasma -- 5.1 Traveling Wave Solutions for Competitive and Monotone Systems -- Part A: Existence of TravelingWave Connecting a Semi-Trivial Steady- State to a Coexistence Steady-State. -- Part B: Iterative Method for obtaining Traveling Wave for General Monotone Systems. -- 5.2 Positive Solutions for Systems of Wave Equations and Their Stabilities -- 5.3 Invariant Manifolds for Coupled Navier-Stokes and Second Order Wave Equations -- Part A: Main Theorem for the Existence of Invariant Manifold. -- Part B: Dependence on Initial Conditions, Asymptotic Stability of the Manifold, and Applications. -- 5.4 Existence and Global Bounds for Fluid Equations of Plasma Display Technology -- 6 Appendices -- 6.1 Existence of Solution between Upper and Lower Solutions for Elliptic and Parabolic Systems, Bifurcation Theorems -- 6.2 The Fixed Point Index, Degree Theory and Spectral Radius of Positive Operators -- 6.3 Theorems Involving Maximum Principle, Comparison and Principal Eigenvalues for Positive Operators -- 6.4 Theorems Involving Derivative Maps, Semigroups and Stability -- 6.5 W2,1 p Estimates, Weak Solutions for Parabolic Equations with Mixed Boundary Data, Theorems Related to Optimal Control, Cross-Diffusion and TravelingWave -- Bibliography -- Index.
Abstract:
The book presents the theory of diffusion-reaction equations starting from the Volterra-Lotka systems developed in the eighties for Dirichlet boundary conditions. It uses the analysis of applicable systems of partial differential equations as a starting point for studying upper-lower solutions, bifurcation, degree theory and other nonlinear methods. It also illustrates the use of semigroup, stability theorems and W 2p theory. Introductory explanations are included in the appendices for non-expert readers. The first chapter covers a wide range of steady-state and stability results involving prey-predator, competing and cooperating species under strong or weak interactions. Many diagrams are included to easily understand the description of the range of parameters for coexistence. The book provides a comprehensive presentation of topics developed by numerous researchers. Large complex systems are introduced for modern research in ecology, medicine and engineering. Chapter 3 combines the theories of earlier chapters with the optimal control of systems involving resource management and fission reactors. This is the first book to present such topics at research level. Chapter 4 considers persistence, cross-diffusion, and boundary induced blow-up, etc. The book also covers traveling or systems of waves, coupled Navier-Stokes and Maxwell systems, and fluid equations of plasma display. These should be of interest to life and physical scientists. Sample Chapter(s). Chapter 1: Positive Solutions for Systems of Two Equations (908 KB). Contents: Positive Solutions for Systems of Two Equations; Positive Solutions for Large Systems of Equations; Optimal Control for Nonlinear Systems of Partial Differential Equations; Persistence, Upper and Lower Estimates, Blowup, Cross-Diffusion and Degeneracy; Traveling Waves, Systems of Waves, Invariant Manifolds, Fluids and
Plasma. Readership: Academic researchers, postgraduates and graduates in differential equations, biomathematics, mathematical modeling, control theory and optimization, mathematicians, engineers, ecologists, biologists, physicists and chemists.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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Electronic Access:
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