Integral manifolds for partial functional differential equations in admissible spaces on a half-line

In this paper we investigate the existence of stable and center-stable manifolds for solutions to partial functional differential equations of the form u ˙ ( t ) = A ( t ) u ( t ) + f ( t , u t ) , t ⩾ 0 , when its linear part, the family of operators ( A ( t ) ) t ⩾ 0 , generates 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 φ-Lipschitz condition, i.e., ‖ f ( t , u t ) − f ( t , v t ) ‖ ⩽ φ ( t ) ‖ u t − v t ‖ C where u t , v t ∈ C : = C ( [ − r , 0 ] , X ) , and φ ( t ) belongs to some admissible function space on the half-line. Our main methods invoke Lyapunov–Perron methods and the use of admissible function spaces.