### 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}. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φ_{t} - Δφ + φ ^{p-1}φ = 0, (t,x) ∈ R_{+} × R^{N}, which has the similarity solution w_{a}(t,x) with the form t^{-1/(p-1)}f(x/√t) depending on a = lim_{ x →∞} x ^{2/(p-1}) f(x) ≥ 0 provided that p is less than the Fujita exponent p_{c}(N) := 1+2/N. In this paper, as a first step, if 1 < p < p_{c}(N) and the data (u_{0}, u_{1}) (x) decays exponentially as x → ∞ without smallness condition, the solution is shown to decay with rates as t → ∞, (∥u(t)∥_{L2}, ∥u(t)∥_{Lp+1}, ∥∇u(t)∥_{L2}) = O (t^{-1/p-1+N/4}, t^{-1/p-1+N/2(p+1)}, t^{-1/p-1-1/2+N/4}), (*) those of which seem to be reasonable, because the similarity solution w_{a} (t,x) has the same decay rates as (*). For the proof, the weighted L^{2}-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

Original language | English |
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Pages (from-to) | 598-610 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 313 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 Jan 15 |

### Keywords

- Decay rate
- Similarity solution
- Subcritical
- Weighted energy method

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*313*(2), 598-610. https://doi.org/10.1016/j.jmaa.2005.08.059