2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142488We study the existence, uniqueness and regularity of solutions of the equation $f_t = Δ_p f = \text{div} (|Df|^{p-2} Df)$ 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, submittedAnalysis of PDEs35R35 (Primary); 35K55,35K65 (Secondary)text