Optimal regularity and stability analysis in the α-Norm for a class of partial functional differential equations with infinite delay
This work aims to investigate the regularity and the stability of the solutions for a class of partial functional differential equations with in finite 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 in finitesimal 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.