Asymptotics of Solutions of Cauchy Problems

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 C0-semigroup T, mild solutions of the homogeneous problem (ACP0) 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.