Lipschitz regularity for parabolic double phase equations with gradient nonlinearity

TL;DR

This paper proves local Lipschitz regularity in space and sharp Hölder continuity in time for solutions to a class of parabolic double phase equations with gradient nonlinearity, using viscosity solution techniques.

Contribution

It introduces new regularity results for parabolic double phase equations with gradient dependence, employing the Ishii-Lions method and establishing solution equivalence under certain conditions.

Findings

Proved local Lipschitz regularity in space for viscosity solutions.

Established sharp Hölder continuity in time.

Showed equivalence between bounded viscosity and weak solutions under regularity assumptions.

Abstract

We establish the local Lipschitz regularity in space for the viscosity solutions to the parabolic double phase equation of the form \[ \smash{\partial_{t}u-\operatorname{div} \left(|Du|^{p-2}D u+a(z)|D u|^{q-2}D u\right)=f(z, Du)} \] by employing the Ishii-Lions method. In addition, we obtain H\"{o}lder estimate in time which turns out to be sharp in the degenerate regime. Here, $1< p\leq q<\infty,$ and the coefficient $a\geq 0$ is assumed to be bounded, locally Lipschitz continuous in space, and continuous in time. Furthermore, the non-homogeneity $f$ is assumed to be continuous on $\Omega\times \mathbb{R}\times \mathbb{R}^N,$ and to satisfy a suitable gradient growth condition. We also establish the equivalence between bounded viscosity solutions and weak solutions, under appropriate additional regularity assumption on the coefficient $a.$