Lipschitz regularity for parabolic double phase equations with gradient nonlinearity

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 $a.$