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

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 (u) over dot (t) = A(t)u(t) + f (t, u(t)), t >= 0. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like Lp -spaces and many other function spaces occurring in interpolation theory such as the Lorentz spaces Lp,q. Results in this paper are the generalization and development for our results in [15]. The existence of these manifolds obtained in the case that the family of operators (A(t)) (t >= 0) generate the evolution family (U(t, s)) t >= s >= 0 having an exponential dichotomy or trichotomy on the half-line and the nonlinear forcing term f satisfies the phi-Lipschitz condition, i.e., parallel to f( t, u(t)) - f (t, v(t))parallel to <= phi(t)parallel to u(t) - v(t)parallel to C, where u(t), v(t) is an element of C := C([-r,0], X), and phi(t) belongs to some admissible Banach function space and satisfies certain conditions.