  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.
