H\"{o}lder continuity of a bounded weak solution of generalized parabolic $p-$Laplacian equations

TL;DR

This paper proves that bounded weak solutions to a generalized class of parabolic p-Laplacian equations are locally Hölder continuous, extending regularity results to broader nonlinear growth conditions within Orlicz space frameworks.

Contribution

It introduces a unified geometric approach to establish Hölder continuity for solutions under diverse growth conditions, generalizing previous results beyond standard p-Laplacian equations.

Findings

Proved local Hölder continuity for solutions with growth between two power functions.

Unified proof applicable to different growth regimes without solution size assumptions.

Extended regularity theory to equations in Orlicz space settings.

Abstract

Here we generalize quasilinear parabolic $p-$Laplacian type equations to obtain the prototype equation as \[ u_t - \text{div} (g(|Du|)/ |Du| \cdot Du) = 0, \] where a nonnegative, increasing, and continuous function $g$ trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $1<g_0 \leq g_1 < \infty$. Through this generalization in the setting from Orlicz spaces, we provide a uniform proof with a single geometric setting that a bounded weak solution is locally H\"{o}lder continuous considering $1 < g_0 \leq g_1 \leq 2$ and $2 \leq g_0 \leq g_1 < \infty$ separately. By using geometric characters, our proof does not rely on any of alternatives which is based on the size of solutions.