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

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.