### Abstract

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}.

Original language | English |
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Pages (from-to) | 937-952 |

Number of pages | 16 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Jan 1 |

Externally published | Yes |

### Keywords

- Nonlinear diffusion equation
- Overdetermined problems
- Stationary level surfaces

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics