### Abstract

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping mathmatical equation repersented we show that the time-global solution of (*) does not exist provided that mathematical equation repersented (Fujita exponent). On the other hand mathematical equation repersented the small data global existence of solution has been recently proved in [K. Nishihara,Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that 0 ≤β < 1. We can prove the small data global existence even if -1 < β < 0. Thus, we conclude that the Fujita exponent ρF (N) is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109-114].

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

Pages (from-to) | 4307-4320 |

Number of pages | 14 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 32 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2012 Dec |

### Fingerprint

### Keywords

- Critical exponent
- Time-dependent damping
- Wave equation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*32*(12), 4307-4320. https://doi.org/10.3934/dcds.2012.32.4307

**Critical exponent for the semilinear wave equation with time-dependent damping.** / Lin, Jiayun; Nishihara, Kenji; Zhai, Jian.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 32, no. 12, pp. 4307-4320. https://doi.org/10.3934/dcds.2012.32.4307

}

TY - JOUR

T1 - Critical exponent for the semilinear wave equation with time-dependent damping

AU - Lin, Jiayun

AU - Nishihara, Kenji

AU - Zhai, Jian

PY - 2012/12

Y1 - 2012/12

N2 - We consider the Cauchy problem for the semilinear wave equation with time-dependent damping mathmatical equation repersented we show that the time-global solution of (*) does not exist provided that mathematical equation repersented (Fujita exponent). On the other hand mathematical equation repersented the small data global existence of solution has been recently proved in [K. Nishihara,Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that 0 ≤β < 1. We can prove the small data global existence even if -1 < β < 0. Thus, we conclude that the Fujita exponent ρF (N) is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109-114].

AB - We consider the Cauchy problem for the semilinear wave equation with time-dependent damping mathmatical equation repersented we show that the time-global solution of (*) does not exist provided that mathematical equation repersented (Fujita exponent). On the other hand mathematical equation repersented the small data global existence of solution has been recently proved in [K. Nishihara,Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that 0 ≤β < 1. We can prove the small data global existence even if -1 < β < 0. Thus, we conclude that the Fujita exponent ρF (N) is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109-114].

KW - Critical exponent

KW - Time-dependent damping

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=84867017613&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867017613&partnerID=8YFLogxK

U2 - 10.3934/dcds.2012.32.4307

DO - 10.3934/dcds.2012.32.4307

M3 - Article

VL - 32

SP - 4307

EP - 4320

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 12

ER -