Polynomial stability and delay semigroups

Abstract

In this paper, we study polynomial stability of the delay semigroup associated with the delay equation
$u^{\prime}(t) = Bu(t) + \Phi u_{t}$, where $u_{t}$ is the history function, $\Phi\in \mathcal{L}(W^{1,p}([-1,0],X),X)$ be a delay operator,
and $(B,D(B))$ is the generator of a polynomially stable semigroup. It is shown that the delay semigroup associated with the delay equation
(DE) generates a polynomially stable semigroup.