Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 115 - 135, Volume 30, Issue 1, May 2011      doi:10.3934/dcds.2011.30.115

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Abdelhai Elazzouzi - Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P.2390 Marrakech, Morocco (email)
Aziz Ouhinou - African Institute for Mathematical Sciences (AIMS), 6 Melrose Road, Muizenberg 7945, South Africa (email)

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.

Keywords:  Analytic semigroup, fractional power of operators, infinite delay, mild and strict solution, uniform fading memory space, essential spectrum, stability.
Mathematics Subject Classification:  Primary: 34K06,34K20; Secondary: 35B35.

Received: January 2010;      Revised: July 2010;      Published: February 2011.