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)| eω t, then the asymptotic stability of the solutions is demonstrated when ω + L < 0. Exact regions of stability are determined for some equations where F is linear.