Theory and Applications of Partial Functional Differential Equations / Edition 1 available in Hardcover
- Pub. Date:
- Springer New York
This book provides an introduction to the qualitative theory and applications of partial functional differential equations from the viewpoint of dynamical systems. Many fundamental results and methods scattered throughout research journals are described, various applications to population growth in a heterogeneous environment are presented and a comprehensive bibliography from both mathematical and biological sources is provided. The main emphasis of the book is on reaction-diffusion equations with delayed nonlinear reaction terms and on the joint effect of the time delay and spatial diffusion on the spatial-temporal patterns of the considered systems.
Table of Contents1. Preliminaries.- 1.1 Semigroups and generators.- 1.2 Function spaces, elliptic operators, and maximal principles.- Bibliographical Notes.- 2. Existence and Compactness of Solution Semiflows.- 2.1 Existence and compactness.- 2.2 Local existence and global continuation.- 2.3 Extensions to neutral partial functional differential equations.- Bibliographical Notes.- 3. Generators and Decomposition of State Spaces for Linear Systems.- 3.1 Infinitesimal generators of solution semiflows of linear systems.- 3.2 Decomposition of state spaces by invariant subspaces.- 3.3 Computation of center, stable, and unstable subspaces.- 3.4 Extensions to equations with infinite delay.- 3.5 L2-stability and reduction of neutral equations.- Bibliographical Notes.- 4. Nonhomogeneous Systems and Linearized Stability.- 4.1 Dual operators and an alternative theorem.- 4.2 Variation of constants formula.- 4.3 Existence of periodic or almost periodic solutions.- 4.4 Principle of linearized stability.- 4.5 Fundamental transformations and representations of solutions.- Bibliographical Notes.- 5. Invariant Manifolds of Nonlinear Systems.- 5.1 Stable and unstable manifolds.- 5.2 Center manifolds.- 5.3 Flows on center manifolds.- 5.4 Global invariant manifolds of perturbed wave equations.- Bibliographical Notes.- 6. Hopf Bifurcations.- 6.1 Some classical Hopf bifurcation theorems for ODEs.- 6.2 Smooth local Hopf bifurcations: a special case.- 6.3 Some examples from population dynamics.- 6.4 Smooth local Hopf bifurcations: general situations.- 6.5 A topological global Hopf bifurcation theory.- 6.6 Global continuation of wave solutions.- Bibliographical Notes.- 7. Small and Large Diffusivity.- 7.1 Destablization of periodic solutions by small diffusivity.- 7.2 Large diffusivity under Neumann boundary conditions.- Bibliographical Notes.- 8. Invariance, Comparison, and Upper and Lower Solutions.- 8.1 Invariance and inequalities.- 8.2 Systems and strict inequalities.- 8.3 Applications to reaction diffusion equations with delay.- Bibliographical Notes.- 9. Convergence, Monotonicity, and Contracting Rectangles.- 9.1 Monotonicity and generic convergence.- 9.2 Stability and steady state solutions of quasimonotone systems.- 9.3 Comparison and convergence results.- 9.4 Applications to Lotka-Volterra competition models.- Bibliographical Notes.- 10. Dispativeness, Exponential Growth, and Invariance Principles.- 10.1 Point dispativeness in a scalar equation.- 10.2 Convergence in a scalar equation.- 10.3 Exponential growth in a scalar equation.- 10.4 An invariance principle.- Bibliographical Notes.- 11. Traveling Wave Solutions.- 11.1 Huxley nonlinearities and phase plane arguments.- 11.2 Delayed Fisher equation: sub-super solution method.- 11.3 Systems and monotone iteration method.- 11.4 Traveling oscillatory waves.- Bibliographical Notes.