Invariant foliations near normally hyperbolic equilibria for quasilinear parabolic problems

TL;DR

This paper proves the existence of stable and unstable foliations near normally hyperbolic equilibria for quasilinear parabolic equations, assuming only C^1-regularity, advancing the understanding of their local dynamics.

Contribution

It establishes the existence of invariant foliations for normally hyperbolic equilibria in quasilinear parabolic problems under minimal regularity assumptions.

Findings

Existence of stable and unstable foliations proven

Foliations constructed under C^1-regularity

Applicable to finite-dimensional manifolds of equilibria

Abstract

We consider quasilinear parabolic evolution equations in the situation where the set of equilibria forms a finite-dimensional C^1-manifold which is normally hyperbolic. The existence of foliations of the stable and unstable manifolds is shown assuming merely C^1-regularity of the underlying equation.