Global behaviour of solutions to a parabolic mean curvature equation

Abstract

Consider (1.1) for a domain $\Omega$ for which there is no classical nonparametric solution of the stationary problem. We study viscosity solutions of (1.1). In general they fail to satisfy Dirichlet data on the boundary and "detach." In fact the solution tends to infinity with finite speed. The velocity stabilizes as $t\to \infty$, and we give some results on asymptotic growth. These new effects can be reconciled with the notion of viscosity solutions. The free boundary data are shown to be Lipschitzian for special domains $\Omega$. Problem (1.1) is related to some isoperimetric geometric problem.