### Abstract

We study the Cauchy problem for the nonlinear heat equation ut-Δu+u1+σ=0,x∈Rn,t>0,u(0,x)=u0(x),x∈Rn, in the sub critical case of σ∈(0,2n). In the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper Escobedo and Kavian (1987) [5]u(t,x)=t-1σw0(xt)+o(t- 1σ) as t→∞ uniformly with respect to x∈Rn, where w0(ξ) is a positive solution of equation -Δw-ξ2·∇w+w1+σ= 1σw which decays rapidly at infinity: lim|ξ|→±∞|ξ |2σw0(ξ)=0.

Original language | English |
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Pages (from-to) | 1585-1595 |

Number of pages | 11 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 74 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2011 Mar 1 |

Externally published | Yes |

### Keywords

- Asymptotics of solutions
- Nonlinear heat equations

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Hayashi, N., & Naumkin, P. I. (2011). Asymptotics for nonlinear heat equations.

*Nonlinear Analysis, Theory, Methods and Applications*,*74*(5), 1585-1595. https://doi.org/10.1016/j.na.2010.10.029