Stable and center-stable manifolds of admissible classes for partial functional differential equations

Abstract

In this paper, we investigate the existence of stable and center-stable manifolds of admissible classes for mild solutions to partial functional differential equations of the form $\dot {u}(t)=A(t)u(t)+f(t,u_t)$, $t\ge 0$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces and many other function spaces occurring in interpolation theory such as the Lorentz spaces $L_{p,q}$. Results in this paper are the generalization and development for our results in \cite {HD1}. The existence of these manifolds obtained in the case that the family of operators $(A(t))_{t\ge 0}$ generate the evolution family $(U(t,s))_{t\ge s\ge 0}$ having an exponential dichotomy or trichotomy on the half-line and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz condition, i.e., $\| f(t,u_t) -f(t,v_t)\| \le \varphi (t)\|u_t -v_t\|_{\mathcal {C}}$, where $u_t,\ v_t \in \mathcal{C} :=C([-r, 0], X)$, and $\varphi (t)$ belongs to some admissible Banach function space and satisfies certain conditions.