Asymptotic behaviour of parabolic problems with delays in the highest order derivatives

We use semigroup methods to investigate the partial functional differential equation u'(t) = Au(t) + integral(-r)(0) dB(theta)u(t + theta) for a sectorial operator A on a Banach space X and a function B : [-r, 0] --> L(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, these properties can be characterized by estimates on (lambda - A - (dB) over cap(lambda))(-1) is an element of L(X, D(A)). Related results on stability also hold for general Banach spaces. The case B = etaA with scalar valued eta is treated in some detail.