TY - JOUR AU - Phi Hà AU - Thiếu Huy Nguyễn AU - Thì Ngọc Hà Vụ PY - 2012/06/25 Y2 - 2024/03/29 TI - Integral manifolds for semilinear evolution equations and admissibility of function spaces JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 64 IS - 6 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2616 AB - We prove the existence of integral (stable, unstable, center) manifolds for the solutions to the semilinear integral equation$u(t) = U(t,s)u(s) + \int^t_s U(t,\xi)f (\xi,u(\xi))d\xi$ in the case where the evolution family $(U(t, s))_{t leq s}$ has an exponentialtrichotomy on a half-line or on the whole line, and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz conditions, i.e., $||f (t, x) — f (t, y) \leq \varphi p(t)||x — y||$, where $\varphi (t)$ 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. ER -