## Abstract

In many cases an inertial manifold 2M for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function Φ from a finite dimensional Hilbert space H^{p} to another Hilbert space H^{q}. The invariance property of ^{M} means that Φ can be written as the solution of a first order partial differential equation DΦ(p)G_{1}(p, Φ(p)) + AΦ(p) = G_{2}(p, Φ(p)) (0) over H^{p}, where G_{1} and G_{2} are nonlinear functions which depend on the original dynamical system and A is a suitably "stable" linear operator. In this paper we use a method introduced by Sacker (R. J. Sacker, A new approach to the perturbation theory of invariant surface, Comm. Pure Appl. Math.18 (1965), 717-732), for the study of finite dimensional dynamical systems, to find inertial manifolds in the infinite dimensional setting. This method involves replacing the first order equation for Φ by the regularized elliptic equation -εΔΦ + DΦ(p) G_{1},(p, Φ(P)) + AΦ(p) = G_{2}(p, Φ(p)), with suitable boundary conditions. It is shown that if A satisfies a spectral gap condition, then the solutions Φ_{ε} of the elliptic equation converge to a weak solution Φ of (0), as ε → 0^{+}. Furthermore, M = Graph Φ is an invariant manifold for the given dynamical system.

Original language | English (US) |
---|---|

Pages (from-to) | 355-387 |

Number of pages | 33 |

Journal | Journal of Differential Equations |

Volume | 89 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1991 |

### Bibliographical note

Funding Information:* This research was supported in part by grants from the National Science Applied Mathematics and Computational Mathematics Program/DARPA, Research Foundation.