An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

TL;DR

This paper develops a new energy formula for fully nonlinear degenerate parabolic equations in one dimension, extending previous methods and providing new tools for analyzing stability and structure of such equations.

Contribution

It introduces a modified method based on Matano's approach to construct energy formulas for fully nonlinear degenerate parabolic equations, including a novel candidate for the porous medium equation.

Findings

Constructed energy formulas for nonlinear degenerate parabolic equations.

Provided examples demonstrating the applicability of the new energy candidate.

Extended the theoretical framework for stability analysis of these equations.

Abstract

Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation.