Integral manifolds for semilinear evolution equations and admissibility of function spaces - Hà, Phi; Nguyễn, Thiếu Huy; Vụ, Thì Ngọc Hà (Ukrains’kyi Matematychnyi Zhurnal, 2012)
We prove the existence of integral (stable, unstable, center) manifolds for the solutions to the semilinear integral equation u(t) = U(t,s)u(s) + ^t_s U(t,ξ)f (ξ,u(ξ))dξ in the case where the evolution family (U(t, s))_t leq s has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the φ -Lipschitz conditions, i.e., ||f (t, x) — f (t, y) ≤ φ p(t)||x — y||, where φ (t) belongs to some classes of admissible function spaces. Our main method invokes the Lyapunov-Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.