### 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 |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

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

**Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption.** / Nishihara, Kenji; Zhao, Huijiang.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 313, no. 2, pp. 598-610. https://doi.org/10.1016/j.jmaa.2005.08.059

}

TY - JOUR

T1 - Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

AU - Nishihara, Kenji

AU - Zhao, Huijiang

PY - 2006/1/15

Y1 - 2006/1/15

N2 - We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + u p-1u = 0, (t,x) ∈ R+ × RN. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φt - Δφ + φ p-1φ = 0, (t,x) ∈ R+ × RN, which has the similarity solution wa(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 pc(N) := 1+2/N. In this paper, as a first step, if 1 < p < pc(N) and the data (u0, u1) (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 wa (t,x) has the same decay rates as (*). For the proof, the weighted L2-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].

AB - We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + u p-1u = 0, (t,x) ∈ R+ × RN. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φt - Δφ + φ p-1φ = 0, (t,x) ∈ R+ × RN, which has the similarity solution wa(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 pc(N) := 1+2/N. In this paper, as a first step, if 1 < p < pc(N) and the data (u0, u1) (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 wa (t,x) has the same decay rates as (*). For the proof, the weighted L2-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].

KW - Decay rate

KW - Similarity solution

KW - Subcritical

KW - Weighted energy method

UR - http://www.scopus.com/inward/record.url?scp=28244468642&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28244468642&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2005.08.059

DO - 10.1016/j.jmaa.2005.08.059

M3 - Article

VL - 313

SP - 598

EP - 610

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -