## Abstract

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation, ∂ _{t}u = Δu+F(x,t,u) in R ^{N} × (0, ∞) u(x, 0) = φ(x) in R ^{N}, (P) where F ∈ C(R ^{N} × [0, ∞) × R) and φ ∈ L ^{1} (R ^{N}, (1 + {pipe}x{pipe}) ^{K} dx) with K ≥ 0. Assume that u is a solution of (P) satisfying {pipe}F(x,t,u(x,t)){pipe} ≤ C(1+t) ^{-A} {pipe}u(x,t){pipe}, (x,t) ∈ R ^{N} × (0,∞) for some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K] is the integer satisfying K - 1 < [K] ≤ K.

Original language | English |
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Pages (from-to) | 161-192 |

Number of pages | 32 |

Journal | Mathematische Annalen |

Volume | 353 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 May 1 |

## ASJC Scopus subject areas

- Mathematics(all)