Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity

TL;DR

This paper develops a regularity theory for a new class of quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity, relevant to stochastic games and double phase problems, establishing local Hölder continuity of spatial gradients.

Contribution

It introduces a novel class of equations combining degeneracy and singularity, and proves local Hölder regularity of their solutions' gradients using geometric methods.

Findings

Established local Hölder regularity of spatial gradients.

Unified treatment of degeneracy and singularity in quasi-linear parabolic equations.

Applicable to models from stochastic games and double phase problems.

Abstract

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D u|}\right\rangle\right), $$ where $1<p<\infty$, $-1<q\leq s<\infty$ and $a(x,t)\ge 0$. The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of $q,s$, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when $q=p-2$ and $q<s$, it will encompass the parabolic $p$-Laplacian both in divergence form and in non-divergence form. We aim to explore the from $L^\infty$ to $C^{1,\alpha}$ regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local H\"{o}lder regularity of spatial gradients of viscosity solutions.