A free-boundary problem for the evolution $p$-Laplacian equation with a combustion boundary condition

Abstract

We study the existence, uniqueness and regularity of solutions of the equation $f_t={\Delta }_{p}f=\text{div}(|Df{|}^{p-2}Df)$ under over-determined boundary conditions $f=0$ and $|Df|=1$. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until it vanishes identically. Furthermore, the free-boundary of the support of $f$ is smooth for all positive time.

25 pages, submitted