### 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 |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis

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## 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