Stable manifolds for some partial neutral functional differential equations with non-dense domain

Abstract

We investigate the asymptotic solution beha\-vior for the partial neutral functional differential equation$$ \begin{cases} & \frac {d}{dt}\mathcal {D}u_t=(A+B(t))\mathcal {D}u_t+f(t,u_t), \quad \hbox {$t\geq s\geq 0$,} \\ & u_s=\phi \in \mathcal {C}:=C([-r,0],X), \end{cases} $$ where the linear operator $A$ is not necessarily densely defined and satisfies the Hille-Yosida condition, and the delayed part $f$ is assumed to satisfy the $\varphi $-Lipschitz condition, i.e., $\|f(t,\phi )-f(t,\psi )\|\leq \varphi (t)\|\phi -\psi \|_{\mathcal {C}}$. Here $\varphi $ belongs to some admissible spaces and $\phi ,\ \psi \in \mathcal {C}:=C([-r,0],X)$.

More precisely, we prove the existence of stable (respectively, center stable) manifolds when the linear part generates an evolution family having an exponential dichotomy (respectively, trichotomy) on the half positive line. Furthermore, we show that such stable manifold attracts all mild solutions of the considered neutral equation. An example is given to assimilate our theory.