# Asymptotics of Solutions of Cauchy Problems

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## Abstract

In this chapter, we give various results concerning the long-time asymptotic behaviour of mild solutions of homogeneous and inhomogeneous Cauchy problems on \(\mathbb {R}_+\) (see Section 3.1 for the definitions and basic properties). For the most part, we shall assume that the homogeneous problem is well-posed, so that the operator *A* generates a *C*_{0}-semigroup *T*, mild solutions of the homogeneous problem (*ACP*0) are given by \(u(t) = T(t)x = :u_x (t) \) (Theorem 3.1.12), and mild solutions of the inhomogeneous problem (*ACPf* ) are given by \(u(t) = T(t)x + (T*f)(t), \) where \( T*f \) is the convolution of *T* and *f* (Proposition 3.1.16). In typical applications, the operator *A* and its spectral properties will be known, but solutions *u* will not be known explicitly, so the objective is to obtain information about the behaviour of *u* from the spectral properties of *A*. To achieve this, we shall apply the results of earlier chapters, making use of the fact that the Laplace transform of *u* can easily be described in terms of the resolvent of *A*.

## Keywords

Cauchy Problem Mild Solution Banach Lattice Strong Operator Topology Kernel Operator## Preview

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