### Abstract

We consider the Cauchy problem for the damped wave equation with absorption{A formula is presented} The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation φ{symbol}_{t} - Δ φ{symbol} + | φ{symbol} |^{ρ - 1} φ{symbol} = 0, ( t, x ) ∈ R_{+} × R^{N}. In the subcritical case 1 < ρ < ρ_{c} ( N ) : = 1 + 2 / N there exists a similarity solution w_{b} ( t, x ) with the form t^{- 1 / ( ρ - 1 )} f ( x / sqrt(t) ) depending on b = lim_{| x | → ∞} | x |^{2 / ( ρ - 1 )} f ( | x | ) {greater than or slanted equal to} 0. Our first aim is to show the decay rates{A formula is presented} provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two- and three-dimensional space. To prove our results, both the weighted L^{2}-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

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

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Journal of Differential Equations |

Volume | 226 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Jul 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*226*(1), 1-29. https://doi.org/10.1016/j.jde.2006.01.002

**Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption.** / Ikehata, Ryo; Nishihara, Kenji; Zhao, Huijiang.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 226, no. 1, pp. 1-29. https://doi.org/10.1016/j.jde.2006.01.002

}

TY - JOUR

T1 - Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

AU - Ikehata, Ryo

AU - Nishihara, Kenji

AU - Zhao, Huijiang

PY - 2006/7/1

Y1 - 2006/7/1

N2 - We consider the Cauchy problem for the damped wave equation with absorption{A formula is presented} The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation φ{symbol}t - Δ φ{symbol} + | φ{symbol} |ρ - 1 φ{symbol} = 0, ( t, x ) ∈ R+ × RN. In the subcritical case 1 < ρ < ρc ( N ) : = 1 + 2 / N there exists a similarity solution wb ( t, x ) with the form t- 1 / ( ρ - 1 ) f ( x / sqrt(t) ) depending on b = lim| x | → ∞ | x |2 / ( ρ - 1 ) f ( | x | ) {greater than or slanted equal to} 0. Our first aim is to show the decay rates{A formula is presented} provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two- and three-dimensional space. To prove our results, both the weighted L2-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

AB - We consider the Cauchy problem for the damped wave equation with absorption{A formula is presented} The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation φ{symbol}t - Δ φ{symbol} + | φ{symbol} |ρ - 1 φ{symbol} = 0, ( t, x ) ∈ R+ × RN. In the subcritical case 1 < ρ < ρc ( N ) : = 1 + 2 / N there exists a similarity solution wb ( t, x ) with the form t- 1 / ( ρ - 1 ) f ( x / sqrt(t) ) depending on b = lim| x | → ∞ | x |2 / ( ρ - 1 ) f ( | x | ) {greater than or slanted equal to} 0. Our first aim is to show the decay rates{A formula is presented} provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two- and three-dimensional space. To prove our results, both the weighted L2-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

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=33646464099&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646464099&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2006.01.002

DO - 10.1016/j.jde.2006.01.002

M3 - Article

AN - SCOPUS:33646464099

VL - 226

SP - 1

EP - 29

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -