# H$\ddot{O}$lder regularity of the parabolic $p$-Laplacian Sylvanus Kupongoh Samaila and Edikan E. Akpanibah

**Abstract:** This paper investigate quasi-linear parabolic $p$-Lapalace equation of the type \begin{equation*} \frac{\partial \varphi}{\partial t} - \text{div}(\frac{h(|\nabla \varphi|)\nabla \varphi}{|\nabla \varphi|})= 0,\end{equation*} under suitable assumptions. We provide a single geometric setting for which a bounded weak solution $\varphi$, is locally H$\ddot{o}$lder continuous in the Sobolve-Orlicz Space $W^1L^{\phi}(\Omega)$ using intrinsic scaling method.

**Keywords: **H$\ddot{o}$lder Continuity, $p$-Laplacian, Sobolev-Orlicz spaces, Intrinsic Scaling Method.

**How to cite this article:**

Sylvanus Kupongoh Samaila and Edikan E. Akpanibah, H$\ddot{o}$lder regularity of the parabolic $p$-Laplacian, International Journal of Advances in Mathematics, Volume 2018, Number 3, Pages 41-53, 2018.