## Abstract

Let Ω be a domain in R^{N}, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂_{t}u = div(|∇u|^{p-2}∇u) and ∂_{t}u = Δu^{m}, where 1 < p < 2 and 0 < m < 1. Let u = u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value is a positive continuous function, or the Cauchy problem where the initial datum equals a nonnegative continuous function multiplied by the characteristic function of the set R^{N}\Ω. Choose an open ball B in Ω whose closure intersects ∂Ω only at one point, and let α > (N+1)(2-p)/2p or α > (N+1)(1-m)/4. Then, we derive asymptotic estimates for the integral of u^{α} over B for short times in terms of principal curvatures of ∂Ω at the point, which tells us about the interaction between fast diffusion and geometry of domain.

Original language | English |
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Pages (from-to) | 680-701 |

Number of pages | 22 |

Journal | Kodai Mathematical Journal |

Volume | 37 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

## Keywords

- Cauchy problem
- Fast diffusion
- Geometry of domain
- Initial behavior
- Initial-boundary value problem
- P-Laplacian
- Porous medium type
- Principal curvatures

## ASJC Scopus subject areas

- Mathematics(all)