Abstract
We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C2. 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 C2-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 RN of some substance whose density is initially a characteristic function of the complement of a domain with bounded C2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SN and the hyperbolic space HN.
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