Hölder continuity of a bounded weak solution of generalized parabolic $p-$Laplacian equations

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 $1<g_0\le g_1<\infty$. 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 Hölder continuous considering $1<g_0\le g_1\le 2$ and $2\le g_0\le g_1<\infty$ separately. By using geometric characters, our proof does not rely on any of alternatives which is based on the size of solutions.