Nonlinear diffusion with a bounded stationary level surface

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C^2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C^2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole R^N of some substance whose density is initially a characteristic function of the complement of a domain with bounded C^2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere S^N and the hyperbolic space H^N.

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