### Abstract

In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |^{ρ - 1} u. Here, b (t, x) = b_{0} (1 + | x |^{2})^{- frac(α, 2)} (1 + t)^{- β} with b_{0} > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L^{2} decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.

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

Pages (from-to) | 403-422 |

Number of pages | 20 |

Journal | Journal of Differential Equations |

Volume | 248 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Jan 15 |

### Fingerprint

### Keywords

- Damped wave equation
- Decay rate
- Global existence
- Weighted energy method

### ASJC Scopus subject areas

- Analysis

### Cite this

^{2}-estimates of solutions for damped wave equations with space-time dependent damping term.

*Journal of Differential Equations*,

*248*(2), 403-422. https://doi.org/10.1016/j.jde.2009.09.022

**L ^{2}-estimates of solutions for damped wave equations with space-time dependent damping term.** / Lin, Jiayun; Nishihara, Kenji; Zhai, Jian.

Research output: Contribution to journal › Article

^{2}-estimates of solutions for damped wave equations with space-time dependent damping term',

*Journal of Differential Equations*, vol. 248, no. 2, pp. 403-422. https://doi.org/10.1016/j.jde.2009.09.022

^{2}-estimates of solutions for damped wave equations with space-time dependent damping term. Journal of Differential Equations. 2010 Jan 15;248(2):403-422. https://doi.org/10.1016/j.jde.2009.09.022

}

TY - JOUR

T1 - L2-estimates of solutions for damped wave equations with space-time dependent damping term

AU - Lin, Jiayun

AU - Nishihara, Kenji

AU - Zhai, Jian

PY - 2010/1/15

Y1 - 2010/1/15

N2 - In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |ρ - 1 u. Here, b (t, x) = b0 (1 + | x |2)- frac(α, 2) (1 + t)- β with b0 > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.

AB - In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |ρ - 1 u. Here, b (t, x) = b0 (1 + | x |2)- frac(α, 2) (1 + t)- β with b0 > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.

KW - Damped wave equation

KW - Decay rate

KW - Global existence

KW - Weighted energy method

UR - http://www.scopus.com/inward/record.url?scp=70349783543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349783543&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2009.09.022

DO - 10.1016/j.jde.2009.09.022

M3 - Article

AN - SCOPUS:70349783543

VL - 248

SP - 403

EP - 422

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -