Nonlinear Systems Of Partial Differential Equations : Applications To Life And Physical Sciences.

Nonlinear Systems Of Partial Differential Equations : Applications To Life And Physical Sciences.

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, bifu...

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Bibliographic Details
Format: eBook
Language:English
Published: World Scientific 2009.
Subjects:
Differential equations, Partial.
Differential equations, Nonlinear.
MATHEMATICS > Differential Equations > General.
Differential equations, Nonlinear
Differential equations, Partial
Nichtlineare partielle Differentialgleichung
Online Access:Click for online access
Table of Contents:
  • Cover13;
  • 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.

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