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Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems

Part of: Topological dynamics Stability theory Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  29 June 2020

Pierre Magal  and
Ousmane Seydi
Pierre Magal*
Affiliation:
Université de Bordeaux, IMB, UMR 5251, F-33076Bordeaux, France and CNRS, IMB, UMR 5251, F-33400Talence, France
Ousmane Seydi
Affiliation:
Département Tronc Commun, Ecole Polytechnique de Thiès, Thiès21001, Sénégal e-mail: oseydi@ept.sn
*
e-mail: pierre.magal@u-bordeaux.fr
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Abstract

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

Keywords

Nonautonomous Cauchy problemnon-densely defined Cauchy problemexponential dichotomy

MSC classification

Primary: 37B55: Nonautonomous dynamical systems
Secondary: 47D62: Integrated semigroups 34D09: Dichotomy, trichotomy
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1347 - 1389
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Copyright
© Canadian Mathematical Society 2020

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