### Abstract

We consider the Cauchy problem for the damped wave equation with absorption u_{tt} - Δu + u_{t} + |u|^{p-1}u = 0, (t,x) ∈ R_{+} × R^{N}, (*) with N = 3,4. The behavior of u as t → ∞ is expected to be the Gauss kernel in the supercritical case ρ > ρa_{c}(N) := 1 + 2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for ρ > 1 + 4/N (N = 1,2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for ρ > ρ_{c}(N)(N = 1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for ρ_{c}(N) < ρ ≤ 1+ 4/N(N = 1,2) and ρ_{c}(N) < p < 1 + 3/N(N = 3). Developing their result, we will show the behavior of solutions for ρ_{c}(N) < p ≤ 1 + 4/N (N = 3), ρ_{c}(N) < ρ < 1 + 4/N (N = 4). For the proof, both the weighted L ^{2}-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5, because the semilinear term is not in C^{2} and the second derivatives are necessary when the explicit formula of solutions is estimated.

Original language | English |
---|---|

Pages (from-to) | 805-836 |

Number of pages | 32 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 58 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 Jul |

Externally published | Yes |

### Fingerprint

### Keywords

- Critical exponent
- Explicit formula
- Global asymptotics
- Semilinear damped wave equation
- Weighted energy method

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Global asymptotics for the damped wave equation with absorption in higher dimensional space.** / Nishihara, Kenji.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 58, no. 3, pp. 805-836. https://doi.org/10.2969/jmsj/1156342039

}

TY - JOUR

T1 - Global asymptotics for the damped wave equation with absorption in higher dimensional space

AU - Nishihara, Kenji

PY - 2006/7

Y1 - 2006/7

N2 - We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + |u|p-1u = 0, (t,x) ∈ R+ × RN, (*) with N = 3,4. The behavior of u as t → ∞ is expected to be the Gauss kernel in the supercritical case ρ > ρac(N) := 1 + 2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for ρ > 1 + 4/N (N = 1,2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for ρ > ρc(N)(N = 1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for ρc(N) < ρ ≤ 1+ 4/N(N = 1,2) and ρc(N) < p < 1 + 3/N(N = 3). Developing their result, we will show the behavior of solutions for ρc(N) < p ≤ 1 + 4/N (N = 3), ρc(N) < ρ < 1 + 4/N (N = 4). For the proof, both the weighted L 2-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5, because the semilinear term is not in C2 and the second derivatives are necessary when the explicit formula of solutions is estimated.

AB - We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + |u|p-1u = 0, (t,x) ∈ R+ × RN, (*) with N = 3,4. The behavior of u as t → ∞ is expected to be the Gauss kernel in the supercritical case ρ > ρac(N) := 1 + 2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for ρ > 1 + 4/N (N = 1,2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for ρ > ρc(N)(N = 1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for ρc(N) < ρ ≤ 1+ 4/N(N = 1,2) and ρc(N) < p < 1 + 3/N(N = 3). Developing their result, we will show the behavior of solutions for ρc(N) < p ≤ 1 + 4/N (N = 3), ρc(N) < ρ < 1 + 4/N (N = 4). For the proof, both the weighted L 2-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5, because the semilinear term is not in C2 and the second derivatives are necessary when the explicit formula of solutions is estimated.

KW - Critical exponent

KW - Explicit formula

KW - Global asymptotics

KW - Semilinear damped wave equation

KW - Weighted energy method

UR - http://www.scopus.com/inward/record.url?scp=33749673748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749673748&partnerID=8YFLogxK

U2 - 10.2969/jmsj/1156342039

DO - 10.2969/jmsj/1156342039

M3 - Article

VL - 58

SP - 805

EP - 836

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -