Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature
Junior da Silva Bessa, João Vitor da Silva, Ginaldo de Santana Sá
TL;DR
We address the interior regularity of bounded viscosity solutions to a class of quasi-linear parabolic equations in non-divergence form with non-homogeneous degeneracy, modeled by ∂_t u = H(x,t,∇u) Δ_p^N u + f in Q_1, where H = |∇u|^p + a(x,t)|∇u|^q and Δ_p^N is the normalized p-Laplacian. The authors develop a geometric tangential methodology, refined oscillation control, and scaling to obtain sharp Hölder regularity of the gradient with exponent α ∈ (0, 1/(1+𝔭)) and time Hölder control, along with optimal growth and non-degeneracy estimates at critical points where ∇u = 0. They also prove a sharp growth rate near local extrema, showing |u(x,t)−u(x0,t0)| ≤ C r^{1+1/(p−1)} in appropriate parabolic cylinders, and establish non-degeneracy via barrier arguments. The results extend non-divergence regularity theory for normalized p-Laplacian models, connect to game-theoretic interpretations, and broaden the understanding of dead-core type phenomena in nonlinear parabolic PDEs with nonstandard growth.
Abstract
In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this class is given by \[ \partial_{t} u = \left(|\nabla u|^{\mathfrak{p}} + \mathfrak{a}(x, t)|\nabla u|^{\mathfrak{q}}\right)Δ_{p}^{\mathrm{N}} u + f(x, t) \quad \text{in} \quad Q_1 = B_1 \times (-1, 0], \] where $p \in (1, \infty)$, $\mathfrak{p}, \mathfrak{q} \in [0, \infty)$, and $\mathfrak{a}, f: Q_1 \to \mathbb{R}$ are suitably defined functions. Our approach is based on geometric tangential methods, incorporating a refined oscillation mechanism, compactness arguments, ``alternative methods,'' and scaling techniques. Furthermore, we derive pointwise estimates in settings exhibiting singular-degenerate or doubly singular signatures. To some extent, our regularity estimates refine and extend previous results from \cite{FZ23} through distinct methodological advancements. Finally, we explore connections between our findings and fundamental nonlinear models in the theory of quasilinear PDEs, which may be of independent interest.