### Abstract

We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|ρ-1u. Here, b(t,x)=b0(1+|x|2)α-2(1+t)-β with b0>0, α,β≥0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N-α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,alpha;,beta;) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

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

Pages (from-to) | 602-614 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 374 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 Feb 15 |

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### Keywords

- Damped wave equation
- Decay rate
- Supercritical case
- The weighted energy method

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*374*(2), 602-614. https://doi.org/10.1016/j.jmaa.2010.09.032

**Decay property of solutions for damped wave equations with space-time dependent damping term.** / Lin, Jiayun; Nishihara, Kenji; Zhai, Jian.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 374, no. 2, pp. 602-614. https://doi.org/10.1016/j.jmaa.2010.09.032

}

TY - JOUR

T1 - Decay property of solutions for damped wave equations with space-time dependent damping term

AU - Lin, Jiayun

AU - Nishihara, Kenji

AU - Zhai, Jian

PY - 2011/2/15

Y1 - 2011/2/15

N2 - We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|ρ-1u. Here, b(t,x)=b0(1+|x|2)α-2(1+t)-β with b0>0, α,β≥0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N-α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,alpha;,beta;) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

AB - We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|ρ-1u. Here, b(t,x)=b0(1+|x|2)α-2(1+t)-β with b0>0, α,β≥0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N-α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,alpha;,beta;) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

KW - Damped wave equation

KW - Decay rate

KW - Supercritical case

KW - The weighted energy method

UR - http://www.scopus.com/inward/record.url?scp=77957799902&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957799902&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2010.09.032

DO - 10.1016/j.jmaa.2010.09.032

M3 - Article

AN - SCOPUS:77957799902

VL - 374

SP - 602

EP - 614

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -