INTEGRAL MANIFOLDS AND THEIR ATTRACTION PROPERTY FOR EVOLUTION EQUATIONS IN ADMISSIBLE FUNCTION SPACES

Abstract

In this paper we investigate the existence of a center-stable manifold for solutions to the semi-linear evolution equation of the form $u(t) = U(t,s) u(s) + \int_s^t U(t,\xi) f(\xi,u(\xi)) d\xi$, $t \ge s \ge 0$, when its linear part, the evolution family $(U(t,s))_{t \ge s \ge 0}$, has an exponential trichotomy on the half-line and the nonlinear forcing term $f$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|f(x)-f(y)\| \le \varphi(t)\|x-y\|$ where $\varphi(t)$ belongs to some class of admissible function spaces on the half-line. Moreover, we consider the existence of unstable manifolds and their attraction property for evolution equations defined on the whole line. Our methods are the Lyapunov-Perron method, the rescaling procedures, and the use of admissible function spaces as in [14, 15].