## Abstract

This paper is concerned with positive solutions of semilinear diffusion equations u_{t} = ε^{2} Δ u + u^{p} in Ω with small diffusion under the Neumann boundary condition, where p > 1 is a constant and Ω is a bounded domain in R^{N} with C^{2} boundary. For the ordinary differential equation u_{t} = u^{p}, the solution u^{0} with positive initial data u^{0} ∈ C(Ω) has a blow-up set S^{0} = (x ∈ Ω|u_{0}(x) = max_{yeΩ} u_{0}(y)) and a blowup profile u^{0}_{*}(x)=(u_{0}(x)^{-(p-1)} - (max_{yeΩ} u_{0}(y))^{-(p-1)})^{-1/(p-1)} outside the blow-up set S^{0}. For the diffusion equation u_{t} = ε^{2} Δ u + u^{p} in Ω under the boundary condition ∂u/∂v = 0 on ∂Ω, it is shown that if a positive function u_{0} ∈ C^{2}(Ω) satisfies ∂u_{0}/∂v = 0 on ∂Ω, then the blow-up profile u^{ε}_{*}(x) of the solution u^{ε} with initial data u_{0} approaches u^{0}_{*}(x) uniformly on compact sets of Ω \ S^{0} as ε → +0.

Original language | English |
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Pages (from-to) | 993-1005 |

Number of pages | 13 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 56 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2004 |

## All Science Journal Classification (ASJC) codes

- General Mathematics