Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains

Abstract

In this paper finite-dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

are constructed. Such results are somewhat surprising because of the continuous spectrum of the linearized equation. These manifolds control the long-time behavior of solutions of these equations and can be used to construct systematic, rigorous expansions of the long-time asymptotics in inverse powers of . They also give a new perspective on the change in the long-time asymptotics of the equation with nonlinear term , when passes through the critical value .