### Abstract

We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term u_{tt}-Δu+b(t)u _{t}+|u|^{ρ-1}u=0, (t,x)∈R_{+}×R ^{N}, (u,u_{t})(0,x)=(u_{0},u_{1})(x), x∈R^{N}. When b(t)=b_{0}(t+1)^{-β} with -1<β<1 and b_{0}>0, we want to seek for the asymptotic profile as t→∞ of the solution u to in the supercritical case ρ>ρ_{F}(N):=1+2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation φ_{t}-1/b(t)Δφ=0, (t,x)∈R_{+}×R^{N}. When N=1, the decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of with source term -1/b(t)(u _{tt}+|u|^{ρ-1}u). Thus, we will show θ _{0}G_{B}(t,x) (θ_{0}: suitable constant) to be an asymptotic profile of u, where G_{B}(t,x) is the fundamental solution of.

Original language | English |
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Pages (from-to) | 185-205 |

Number of pages | 21 |

Journal | Asymptotic Analysis |

Volume | 71 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2011 |

### Keywords

- asymptotic profile
- supercritical exponent
- time-dependent damping
- wave equation

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Asymptotic Analysis*,

*71*(4), 185-205. https://doi.org/10.3233/ASY-2010-1018