LAUSR

For the following class of partial neutral functional differential equations  ∂ ∂tFut = B(t)u(t) + Φ(t, ut) t ∈ (0,∞), u0 = φ ∈ C := C([−r, 0], X) we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of E-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators (B(t))t≥0 generates the evolution family {U(t, s)}t≥s≥0 (on Banach space X) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator Φ satisfies the φ-Lipschitz condition, i.e., ‖Φ(t, φ)−Φ(t, ψ)‖ ≤ φ(t)‖φ− ψ‖C for φ, ψ ∈ C, where φ(t) belongs to some admissible function space on the halfline. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.