Optimal regularity and stability analysis
in the $\alpha-$Norm for a class of partial functional
differential equations with infinite delay
Abdelhai Elazzouzi - Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P.2390 Marrakech, Morocco (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.
Received: January 2010; Revised: July 2010; Available Online: February 2011.
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