### Abstract

We consider the Cauchy problem for the damped wave equationu_{t t} - Δ u + u_{t} = | u |^{ρ - 1} u, (t, x) ∈ R_{+} × R^{N} and the heat equationφ{symbol}_{t} - Δ φ{symbol} = | φ{symbol} |^{ρ - 1} φ{symbol}, (t, x) ∈ R_{+} × R^{N} . If the data is small and slowly decays likely c_{1} (1 + | x |)^{- k N}, 0 < k ≤ 1, then the critical exponent is ρ_{c} (k) = 1 + frac(2, k N) for the semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space R^{N}, whose asymptotic profile is given byΦ_{0} (t, x) = under(∫, R^{N}) frac(e^{- frac(| x - y |2, 4 t)}, (4 π t)^{N / 2}) frac(c_{1}, (1 + | y |^{2})^{k N / 2}) d y provided that the data φ{symbol}_{0} satisfies lim_{| x | → ∞} 〈 x 〉^{k N} φ{symbol}_{0} (x) = c_{1}(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, u_{t}) (0, x) = (u_{0}, u_{1}) (x) is shown in low dimensional spaces R^{N}, N = 1, 2, 3, to have the same asymptotic profile Φ_{0} (t, x) provided that lim_{| x | → ∞} 〈 x 〉^{k N} (u_{0} + u_{1}) (x) = c_{1}(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.

Original language | English |
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Pages (from-to) | 803-819 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 338 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 Feb 15 |

### Keywords

- Asymptotic profile
- Damped wave equation
- Slowly decaying data

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*338*(2), 803-819. https://doi.org/10.1016/j.jmaa.2007.05.068