Skip to main content
main-content

Über dieses Buch

Abstract semilinear functional differential equations arise from many biological, chemical, and physical systems which are characterized by both spatial and temporal variables and exhibit various spatio-temporal patterns. The aim of this book is to provide an introduction of the qualitative theory and applications of these equations from the dynamical systems point of view. The required prerequisites for that book are at a level of a graduate student. The style of presentation will be appealing to people trained and interested in qualitative theory of ordinary and functional differential equations.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
The purpose of this introduction is twofold: to present some examples of abstract semilinear functional differential equations and to describe the organization of the book.
Jianhong Wu

1. Preliminaries

Abstract
The purpose of this chapter is to collect some background materials required throughout this book. These materials include semigroup theory, Sobolev spaces, and elliptic operators. We will only state results and leave the details to the references listed in the bibliographical notes.
Jianhong Wu

2. Existence and Compactness of Solution Semiflows

Abstract
The purpose of this chapter is to establish the existence and compactness of solution semiflows defined by a class of semilinear functional differential equations. We will start with the case where the linear operator generates a C 0-semigroup of bounded linear operators and the nonlinear term satisfies a global Lipschitz condition. We will then indicate how to relax this global Lipschitz condition by imposing compactness on the linear semigroup. In the final section, we will derive a class of semilinear functional differential equations of neutral type from a continuous array of coupled lossless transmission lines and we will obtain the basic existence-uniqueness result for such a class of equations.
Jianhong Wu

3. Generators and Decomposition of State Spaces for Linear Systems

Abstract
The purpose of this chapter is to study the strongly continuous semiflow of bounded operators defined by abstract linear functional differential equations. Of main interest are the infinitesimal generators, their spectral properties, the distribution of characteristic values, the decomposition of state spaces according to characteristic values, and the computation of center subspaces. These results will be applied in later chapters to establish stable, unstable, and center manifold theory as well as the Hopf bifurcation theory for nonlinear systems.
Jianhong Wu

4. Nonhomogeneous Systems and Linearized Stability

Abstract
This chapter deals with linear nonhomogeneous systems and linearized stability.
Jianhong Wu

5. Invariant Manifolds of Nonlinear Systems

Abstract
The goal of this chapter is to prove various invariant manifold theorems and to demonstrate their applications to the study of saddle-point property and finite dimensional reduction near centers.
Jianhong Wu

6. Hopf Bifurcations

Abstract
In this chapter, we develop a Hopf bifurcation theory for similinear functional differential equations.
Jianhong Wu

7. Small and Large Diffusivity

Abstract
The central subject of this chapter is the effect of small/large diffusivity on the asymptotic behaviors of solutions to reaction diffusion equations with delay.
Jianhong Wu

8. Invariance, Comparison, and Upper and Lower Solutions

Abstract
This chapter establishes invariance and various inequalities for solutions of abstract integral equations with delay.
Jianhong Wu

9. Convergence, Monotonicity, and Contracting Rectangles

Abstract
This chapter establishes various results on monotonicity, convergence, and stability for reaction diffusion equations with delay subject to Neumann boundary conditions.
Jianhong Wu

10. Dispativeness, Exponential Growth, and Invariance Principles

Abstract
The first part of this chapter is devoted to the scalar equation
$$ \frac{{\partial u\left( {x,t} \right)}}{{\partial t}} = \mu \Delta u\left( {x,t} \right) + f\left( {u\left( {x,t} \right),u\left( {x,t - r} \right)} \right),\quad t > 0,x \in \Omega \subset {\mathbb{R}^N} $$
motivated by Hutchinson’s population model. In Section 10.1, a reduction technique due to Luckhaus will be introduced and applied to establish the point dispativeness of the solution semiflow. It will be shown that, under a negative feedback condition on f, the solution semiflow is point dissipative in one space dimension for arbitrary delay and diffusion coefficients. In higher space dimensions, the point dispativeness still holds provided either the delay is small or the diffusion coefficient is large. But this dispativeness fails to hold if the diffusion coefficient is small and the delay is large. Indeed, as will be proved in Section 10.3, unbounded (exponentially growing) solutions exist in the case of large delay and small diffusion. This indicates that for small value of the diffusion coefficient, spatial heterogeneities no longer stay bounded for large t; rather, small irregularities in the initial distribution tend to grow larger and larger during the time evolution. Section 10.2 will be devoted to a convergence result for the above scalar equation by employing a naturally motivated Liapunov function. The second part of this chapter will develop a general invariance principle of Liapunov-Razumikhin type and its application.
Jianhong Wu

11. Traveling Wave Solutions

Abstract
This chapter deals with the existence of wave solutions in a system where spatial diffusion and temporal delay play a crucial role in determining the system’s spatio-temporal patterns and dynamics.
Jianhong Wu

Backmatter

Weitere Informationen