The exponential dichotomy and invariant manifolds for some classes of differential equations

Abstract: We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other mild assumptions, we investigate the existence, persistence and regularity of different types of invariant manifolds for these differential equations based on our previous works about invariant manifold theory for abstract `generalized dynamical systems': invariant graphs (global version) and normally hyperbolic invariant manifolds (local version); brief summaries of those works are also given.