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

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U2 - 10.3934/dcds.2012.32.4307

DO - 10.3934/dcds.2012.32.4307

M3 - Article

AN - SCOPUS:84867017613

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 -