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

## TL;DR

This paper investigates the existence and properties of invariant manifolds in certain semi-linear differential equations, including ill-posed cases, under exponential dichotomy conditions, extending previous invariant manifold theory.

## Contribution

It extends invariant manifold theory to semi-linear differential equations with exponential dichotomy, covering both well-posed and ill-posed cases, and discusses their persistence and regularity.

## Key findings

- Invariant manifolds exist under exponential dichotomy.
- Persistence and regularity of invariant manifolds are established.
- Applicable to both well-posed and ill-posed 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.