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Asymptotics of Solutions of Cauchy Problems

  • Wolfgang ArendtEmail author
  • Charles J. K. Batty
  • Matthias Hieber
  • Frank Neubrander
Chapter
  • 1.5k Downloads
Part of the Monographs in Mathematics book series (MMA, volume 96)

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.

Keywords

Cauchy Problem Mild Solution Banach Lattice Strong Operator Topology Kernel Operator 
These 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.

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Wolfgang Arendt
    • 1
    Email author
  • Charles J. K. Batty
    • 2
  • Matthias Hieber
    • 3
  • Frank Neubrander
    • 4
  1. 1.Angewandte AnalysisUniversität UlmUlmGermany
  2. 2.St. John’s CollegeOxfordUK
  3. 3.Fachbereich MathematikTU DarmstadtDarmstadtGermany
  4. 4.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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