Gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure

TL;DR

This paper proves Lipschitz and Hölder regularity for viscosity solutions of a class of quasilinear parabolic equations with mixed singular-degenerate structure, extending previous homogeneous case results.

Contribution

It introduces a new approach combining Jensen–Ishii and intrinsic scaling methods to establish regularity for equations with nonhomogeneous degeneracy or singularity.

Findings

Established Lipschitz regularity for solutions.

Proved interior Hölder continuity of the gradient.

Extended regularity results to nonhomogeneous degenerate equations.

Abstract

We establish regularity results for viscosity solutions to a class of quasilinear parabolic equations exhibiting nonhomogeneous degeneracy or singularity (a double phase regime) of the form \[ u_t - \big(|Du|^{\mathfrak{p}} + \mathfrak{a}(x,t)|Du|^{\mathfrak{q}}\big)\Delta_p^{\mathrm N} u = f(x,t) \quad \text{in } Q_1, \] where $-1 < \mathfrak{p} < 0$, $\mathfrak{p} \leq \mathfrak{q}$, and $\mathfrak{a}, f : Q_1 \to \mathbb{R}$ are prescribed functions. Using the Jensen--Ishii method, we prove Lipschitz regularity for appropriately translated solutions. Moreover, combining this approach with intrinsic scaling techniques, we establish interior H\"older continuity estimates for the gradient. Our results extend recent work of Fang and Zhang on the homogeneous case via a different approach.