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

Abstract

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,ut), 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,ut)−f(t,vt)‖⩽φ(t)‖ut−vt‖C where ut,vt∈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.