Longtime Behavior for a Class of Abstract Integrodifferential Equations

Abstract

We are concerned with semilinear integrodifferential equations of the form,

$$ \dot x\left( \psi \right)\left( {\text{t}} \right) = {\text{AX}}\left( \psi \right)\left( {\text{t}} \right) + \int_{ - \infty }^{\text{t}} {{\text{g}}\left( {{\text{t,s,x}}\left( \psi \right)\left( {\text{s}} \right)} \right){\text{ds}}} $$

(1.a)

$$ {\text{x}}\left( \psi \right)\left( \theta \right) = \psi \left( \theta \right)\quad \theta \in \left( { - \infty ,0} \right] $$

(1.b)

$$ \psi \in {\text{C}}_\alpha \quad {\text{For some }}\alpha \in \left( {0,1} \right) $$

(1.c)

Here A is required to be the infinitesional generator of an analytic semi-group {T(t)|t ≥ 0} acting on a Banach space $\{ T(t)|t \geq 0\}$. We further stipulate that O ∈ ρ(A). For α∈(0, 1), A^α denotes the fractional power of the operator A and $\underline {\bar X} _\alpha$ represents the interpolation space defined by the α power of A, i.e.

$$ \underline {\bar X} _\alpha = \{ x|x \in D(A\alpha )\} $$

with

$$ \left\| {\text{x}} \right\|_\alpha = \left\| {{\text{A}}^\alpha {\text{x}}} \right\|. $$