Finite-dimensional reduction of systems of nonlinear diffusion equations

TL;DR

This paper introduces a class of nonlinear diffusion systems where long-term phase behavior can be effectively captured by a finite-dimensional ODE, expanding the known conditions beyond periodic cases.

Contribution

It provides new sufficient conditions for finite-dimensional reduction in nonlinear parabolic systems with Dirichlet boundary conditions, broader than those for periodic cases.

Findings

Long-time phase dynamics described by Lipschitz ODEs in R^n.

Finite-dimensional reduction conditions are wider for Dirichlet problems.

Applicable to a class of nonlinear parabolic equations.

Abstract

We present a class of one-dimensional systems of nonlinear parabolic equations for which long-time phase dynamics can be described by an ODE with a Lipschitz vector field in R^n. In the considered case of the Dirichlet boundary value problem sufficient conditions for a finite-dimensional reduction turn out to be much wider than the known conditions of this kind for a periodic situation.