An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle

TL;DR

This paper develops an explicit Lyapunov function for scalar parabolic PDEs with reflection symmetry on the circle, extending previous methods to periodic boundary conditions.

Contribution

It introduces a novel explicit Lyapunov function for symmetric parabolic PDEs on the circle, adapting classical methods to periodic boundary conditions.

Findings

Constructed an explicit Lyapunov function for symmetric parabolic PDEs.

Extended classical Lyapunov methods to periodic boundary conditions.

Provides a tool for analyzing stability of symmetric PDEs on the circle.

Abstract

We construct an explicit Lyapunov function for scalar parabolic reaction-advection-diffusion equations under periodic boundary conditions. We assume the nonlinearity is even in the advection term. We follow a method originally suggested by Matano and Zelenyak for, and limited to, separated boundary conditions.