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 <!-- MATH $du(t)/dt = Au(t) + F({u_t})$ --> , where is the infinitesimal generator of a strongly continuous semigroup of linear operators <!-- MATH $T(t),t \geqslant 0$ --> , on a Banach space and is a Lipschitz operator from <!-- MATH $C = C([ - r,0];X)$ --> to . The solutions are studied as a semigroup of linear or nonlinear operators on . In the case that has Lipschitz constant and <!-- MATH $|T(t)| \leqslant {e^{\omega t}}$ --> , then the asymptotic stability of the solutions is demonstrated when <!-- MATH $\omega + L < 0$ --> <img width="93" height="38" align="MIDDLE" border="0" src="images/img12.gif" alt="$ \omega + L < 0$">. Exact regions of stability are determined for some equations where is linear.