Optimal regularity and stability analysisin the $\alpha-$Norm for a class of partial functionaldifferential equations with infinite delay

Abdelhai Elazzouzi; Aziz Ouhinou

doi:10.3934/dcds.2011.30.115

Article Contents

2011, Volume 30, Issue 1: 115-135. Doi: 10.3934/dcds.2011.30.115

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Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay

Abdelhai Elazzouzi^1, and
Aziz Ouhinou^2,

1.
Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P.2390 Marrakech
2.
African Institute for Mathematical Sciences (AIMS), 6 Melrose Road, Muizenberg 7945

Received: January 2010

Revised: July 2010

Published: January 2011

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Abstract

This work aims to investigate the regularity and the stability of the solutions for a class of partial functional differential equations with infinite delay. Here we suppose that the undelayed part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator. First, we give a new characterization for the infinitesimal generator of the solution semigroup, which allows us to give necessary and sufficient conditions for the regularity of solutions. Second, we investigate the stability of the semigroup solution. We proved that one of the fundamental and wildly used assumption, in the computing of eigenvalues and eigenvectors, is an immediate consequence of the already considered ones. Finally, we discuss the asymptotic behavior of solutions.

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Mathematics Subject Classification: Primary: 34K06,34K20; Secondary: 35B35.

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References

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