Some invariant manifolds for abstract functional differential equations and linearized stabilities (Q1411117)

The paper is concerned with the abstract functional-differential equation \[ u^{\prime}(t)=Au(t)+L(u_{t})+f(u_{t}),\tag{e} \] where \(A\) is the infinitesimal generator of a strongly continuous compact semigroup on a Banach space \(X,\) \(u_{t}(\theta)=u(t+\theta)\) for \(\theta \in(-\infty,0],\) \(L:\mathcal{B}\rightarrow X\) is a bounded linear operator, \(f\in C^{1}(\mathcal{B},X),\) and \(\mathcal{B}=\mathcal{B}((-\infty,0],X)\) is the fading memory space which satisfies certain axioms. Using the variation of constants formula established recently by \textit{Y. Hino, S. Murakami, T.Naito} and \textit{Nguyen Van Minh} [J. Differ. Equ. 179, 336--355 (2002; Zbl 1005.34070)], the authors establish the existence of local stable, unstable, and center-unstable manifolds for equation (e). As a consequence of these results, criteria for stability and instability of the trivial solution of the equation under consideration are derived. As an application, the stability of the trivial solution for a partial integro-differential equation is discussed.