Existence and stability for partial functional differential equations

The existence and stability properties of a class of partial functional differential equations are investigated. The problem is formulated as an abstract ordinary functional differential equation of the form du (t) /dt = Au (t) + F (uₜ), where A is the infinitesimal generator of a strongly continuous semigroup of linear operators T (t), t 0, on a Banach space X and F is a Lipschitz operator from C = C (- r, 0;X) to X. The solutions are studied as a semigroup of linear or nonlinear operators on C. In the case that F has Lipschitz constant L and |T (t) | e^{ t}, then the asymptotic stability of the solutions is demonstrated when + L < 0. Exact regions of stability are determined for some equations where F is linear.