Center manifolds for infinite dimensional nonautonomous differential equations

[摘要] We study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions. We assume that the linear part of the equation admits, in a very general sense, a decomposition into center and hyperbolic parts. The center manifold is obtained directly as the graph of a fixed point for a Lyapunov-Perron type integral operator. We prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator Lambda. The operator Lambda is formed by the Green's function for the hyperbolic part and composition operators induced by a now on the center part. We formulate the usual gap condition in spectral terms and show that this condition is, in fact, a condition of boundedness of ii on corresponding spaces of differentiable functions. This gives a direct proof of the existence of a smooth global center manifold. (C) 1997 Academic Press.

[发布日期] 1997-12-10