On the Dynamics of Abstract Retarded Evolution Equations

Abstract

This paper is concerned with the dynamics of the following abstract retarded evolution equation: $\left(d/dt\right)$ $u\left(t\right)+Au\left(t\right)=F\left(u\left(t-r_1\right),\dots ,u\left(t-r_n\right)\right)+g\left(t\right)$ in a Hilbert space $H$, where $A:D\left(A\right)\subset H\to H$ is a self-adjoint positive-definite operator with compact resolvent and $F:D{\left(A^{\alpha }\right)}^n\to H$ $\left(\alpha \in \left[0,1/2\right]\right)$ is a locally Lipschitz continuous mapping. The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.