A Lyapunov function for fully nonlinear parabolic equations in one spatial variable

TL;DR

This paper develops a Lyapunov function for fully nonlinear parabolic equations in one spatial dimension, extending previous methods to more complex nonlinear boundary conditions.

Contribution

It introduces a modified approach based on Matano's method to construct Lyapunov functions for fully nonlinear parabolic equations with Robin boundary conditions.

Findings

Successfully constructs Lyapunov functions for fully nonlinear equations

Extends stability analysis techniques to more general boundary conditions

Provides a framework for analyzing nonlinear parabolic PDEs

Abstract

Lyapunov functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano's method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.