Hölder continuity of the gradient of the solutions of certain degenerate parabolic equations (Q1114078)

This paper is concerned with the parabolic equation \[ \partial u/\partial t-div(| \nabla u|^{p-2}\nabla u)=0,\quad x\in \Omega \subset {\mathbb{R}}^ N,\quad t>0 \] with \(p>\max \{3/2,2N/(N+2)\}\) which is degenerate if \(p<2\) or singular if \(3/2<p<2\). Let u(x,t) be any weak solution of the equation in \(L^{\infty}[0,T;L^ 2(\Omega)]\cap L^ p[0,T;W^{1,p}(\Omega)].\) The Hölder continuity of \(\nabla u\) is established.