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 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 o-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 (ACP f ) 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.KeywordsHilbert SpaceCauchy ProblemMild SolutionBanach LatticeStrong Operator TopologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.