Asymptotic Behaviour of Parabolic Problems with Delays in the Highest Order Derivatives

Abstract

We use semigroup methods to investigate the partial functional differential equation $u’(t)=Au(t)+ \int_{-r}^0 dB(\theta)u(t+\theta)$ for a sectorial operator $A$ on a Banach space $X$ and a function $B:[-r,0] \to\cL(D(A),X)$ of bounded variation having no mass at 0. Using a perturbation theorem due to Weiss and Staffans, we construct the solution semigroup on a product space in order to solve the delay equation in a classical sense. Employing the spectrum of the semigroup and its generator, we then study exponential dichotomy and stability of solutions. If $X$ is a Hilbert space, %C% these properties can be characterized by estimates on $(\la-A-\widehat{dB}(\la))^{-1}\in\cL(X,D(A))$. Related results on stability also hold for general Banach spaces. The case $B=\eta A$ with scalar valued $\eta$ is treated in some detail.