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Über dieses Buch

Several types of differential equations, such as functional differential equation, age-structured models, transport equations, reaction-diffusion equations, and partial differential equations with delay, can be formulated as abstract Cauchy problems with non-dense domain. This monograph provides a self-contained and comprehensive presentation of the fundamental theory of non-densely defined semilinear Cauchy problems and their applications. Starting from the classical Hille-Yosida theorem, semigroup method, and spectral theory, this monograph introduces the abstract Cauchy problems with non-dense domain, integrated semigroups, the existence of integrated solutions, positivity of solutions, Lipschitz perturbation, differentiability of solutions with respect to the state variable, and time differentiability of solutions. Combining the functional analysis method and bifurcation approach in dynamical systems, then the nonlinear dynamics such as the stability of equilibria, center manifold theory, Hopf bifurcation, and normal form theory are established for abstract Cauchy problems with non-dense domain. Finally applications to functional differential equations, age-structured models, and parabolic equations are presented. This monograph will be very valuable for graduate students and researchers in the fields of abstract Cauchy problems, infinite dimensional dynamical systems, and their applications in biological, chemical, medical, and physical problems.


Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

The goal of this chapter is to introduce some fundamental theories for Ordinary Differential Equations (ODEs), Retarded Functional Differential Equations (RFDEs), and Age-structured Models and to derive abstract semilinear Cauchy problems from these equations. It serves two purposes: to present a brief review of the basic results on the nonlinear dynamics of these three types of equations and to give a quick preview about the types of results we will develop for the abstract semilinear Cauchy problems in this monograph.
Pierre Magal, Shigui Ruan

Chapter 2. Semigroups and Hille-Yosida Theorem

The aim of this chapter is to introduce the basic concepts and results about semigroups, resolvents, and infinitesimal generators for linear operators and to present the Hille-Yosida theorem for strongly continuous semigroups.
Pierre Magal, Shigui Ruan

Chapter 3. Integrated Semigroups and Cauchy Problems with Non-dense Domain

The goal of this chapter is to introduce the integrated semigroup theory and use it to investigate the existence and uniqueness of integrated (mild) solutions of the nonhomogeneous Cauchy problems when the domain of the linear operator A is not dense in the state space and A is not a Hille-Yosida operator.
Pierre Magal, Shigui Ruan

Chapter 4. Spectral Theory for Linear Operators

This chapter covers fundamental results on the spectral theory, including Fredholm alternative theorem and Nussbaum’s theorem on the radius of essential spectrum for bounded linear operators; growth bound and essential growth bound of linear operators; the relationship between the spectrum of semigroups and the spectrum of their infinitesimal generators; spectral decomposition of the state space; and asynchronous exponential growth of linear operators. The estimates of growth bound and essential growth bound of linear operators will be used in proving the center manifold theorem in Chapter 6
Pierre Magal, Shigui Ruan

Chapter 5. Semilinear Cauchy Problems with Non-dense Domain

The main purpose of this chapter is to present a comprehensive semilinear theory that will allow us to study the properties of solutions of the non-densely defined Cauchy problems, such as existence and uniqueness of a maximal semiflow, positivity, Lipschitz perturbation, differentiability with respect to the state variable, time differentiability, classical solutions, stability of equilibria, etc.
Pierre Magal, Shigui Ruan

Chapter 6. Center Manifolds, Hopf Bifurcation, and Normal Forms

The purpose of this chapter is to develop the center manifold theory, Hopf bifurcation theorem, and normal form theory for abstract semilinear Cauchy problems with non-dense domain.
Pierre Magal, Shigui Ruan

Chapter 7. Functional Differential Equations

The goal of this chapter is to apply the theories developed in the previous chapters to functional differential equations. In Section 7.1 retarded functional differential equations are rewritten as abstract Cauchy problems and the integrated semigroup theory is used to study the existence of integrated solutions and to establish a general Hopf bifurcation theorem. Section 7.2 deals with neutral functional differential equations. In Section 7.3, firstly it is shown that a delayed transport equation for cell growth and division has asynchronous exponential growth; secondly it is demonstrated that partial functional differential equations can also be set up as abstract Cauchy problems.
Pierre Magal, Shigui Ruan

Chapter 8. Age-Structured Models

In this chapter we apply the results obtained in the previous chapters to age-structured models. In Section 8.1, a Hopf bifurcation theorem is established for the general age-structured systems. Section 8.2 deals with a susceptible-infectious epidemic model with age of infection, uniform persistence of the model is established, local and global stability of the disease-free equilibrium is studied by spectral analysis, and global stability of the unique endemic equilibrium is discussed by constructing a Liapunov functional. Section 8.3 focuses on a scalar age-structured model, detailed results on the existence of integrated solutions, local stability of equilibria, Hopf bifurcation, and normal forms are presented.
Pierre Magal, Shigui Ruan

Chapter 9. Parabolic Equations

The theories developed in the previous chapters can be used to study some parabolic equations as well. In this chapter, we first consider linear abstract Cauchy problems with non-densely defined and almost sectorial operators; that is, the part of this operator in the closure of its domain is sectorial. Such problems naturally arise for parabolic equations with nonhomogeneous boundary conditions. By using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem. Then in the second section we provide detailed stability and bifurcation analyses for a scalar reaction-diffusion equation, namely, a size-structured model.
Pierre Magal, Shigui Ruan

Backmatter

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