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

摘要

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., {norm of matrix}f(t,u_t)-f(t,v_t){norm of matrix}≤φ(t){norm of matrix}u_t-v_t{norm of matrix}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.