Interaction between nonlinear diffusion and mean curvature of a surface with initial discontinuities

We consider the Cauchy problems for a class of nonlinear diffusion equations where each initial data is given by a characteristic function of an open set $Ω$ in the whole Euclidean space. If the boundary $\partialΩ$ is of class of $C^2$ in a neighborhood of a point $x$ on it, then we extract the mea