Quantitative stability for quasilinear parabolic equations

TL;DR

This paper investigates the stability of viscosity solutions to quasilinear parabolic PDEs under perturbations, providing explicit convergence rates even in singular or degenerate cases, including p-parabolic equations.

Contribution

It introduces a framework that yields quantitative stability estimates for a broad class of quasilinear parabolic equations, covering perturbations in the exponent and regularized limits.

Findings

Established explicit convergence rates for viscosity solutions

Covered both normalized and variational p-parabolic equations

Provided quantitative estimates for perturbations of the exponent p

Abstract

We examine the stability of a class of quasilinear parabolic partial differential equations under perturbations. We are interested in the behavior of viscosity solutions as the perturbation parameter vanishes and establish explicit convergence rates by adapting standard comparison arguments. Despite the possible singular or degenerate nature of the parabolic operator, our framework covers, in particular, both the normalized and the variational $p$-parabolic equations, providing quantitative estimates for perturbations of the exponent $p$ and limits arising from regularized approximations.