Existence and Stability for Partial Functional Differential Equations

Abstract

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 $du(t)/dt = Au(t) + F({u_t})$, where $A$ is the infinitesimal generator of a strongly continuous semigroup of linear operators $T(t),t \geqslant 0$, 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 $|T(t)| \leqslant {e^{\omega t}}$, then the asymptotic stability of the solutions is demonstrated when $\omega + L < 0$. Exact regions of stability are determined for some equations where $F$ is linear.