### 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 |
---|---|

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 |

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### Keywords

- Asymptotic profile
- Damped wave equation
- Slowly decaying data

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

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

**Asymptotic behavior of solutions for the damped wave equation with slowly decaying data.** / Narazaki, Takashi; Nishihara, Kenji.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 338, no. 2, pp. 803-819. https://doi.org/10.1016/j.jmaa.2007.05.068

}

TY - JOUR

T1 - Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

AU - Narazaki, Takashi

AU - Nishihara, Kenji

PY - 2008/2/15

Y1 - 2008/2/15

N2 - We consider the Cauchy problem for the damped wave equationut t - Δ u + ut = | u |ρ - 1 u, (t, x) ∈ R+ × RN and the heat equationφ{symbol}t - Δ φ{symbol} = | φ{symbol} |ρ - 1 φ{symbol}, (t, x) ∈ R+ × RN . If the data is small and slowly decays likely c1 (1 + | x |)- k N, 0 < k ≤ 1, then the critical exponent is ρc (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 RN, whose asymptotic profile is given byΦ0 (t, x) = under(∫, RN) frac(e- frac(| x - y |2, 4 t), (4 π t)N / 2) frac(c1, (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) = c1(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, ut) (0, x) = (u0, u1) (x) is shown in low dimensional spaces RN, N = 1, 2, 3, to have the same asymptotic profile Φ0 (t, x) provided that lim| x | → ∞ 〈 x 〉k N (u0 + u1) (x) = c1(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.

AB - We consider the Cauchy problem for the damped wave equationut t - Δ u + ut = | u |ρ - 1 u, (t, x) ∈ R+ × RN and the heat equationφ{symbol}t - Δ φ{symbol} = | φ{symbol} |ρ - 1 φ{symbol}, (t, x) ∈ R+ × RN . If the data is small and slowly decays likely c1 (1 + | x |)- k N, 0 < k ≤ 1, then the critical exponent is ρc (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 RN, whose asymptotic profile is given byΦ0 (t, x) = under(∫, RN) frac(e- frac(| x - y |2, 4 t), (4 π t)N / 2) frac(c1, (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) = c1(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, ut) (0, x) = (u0, u1) (x) is shown in low dimensional spaces RN, N = 1, 2, 3, to have the same asymptotic profile Φ0 (t, x) provided that lim| x | → ∞ 〈 x 〉k N (u0 + u1) (x) = c1(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.

KW - Asymptotic profile

KW - Damped wave equation

KW - Slowly decaying data

UR - http://www.scopus.com/inward/record.url?scp=34848897850&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34848897850&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2007.05.068

DO - 10.1016/j.jmaa.2007.05.068

M3 - Article

AN - SCOPUS:34848897850

VL - 338

SP - 803

EP - 819

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -