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

Jiayun Lin, Kenji Nishihara, Jian Zhai

    Research output: Contribution to journalArticle

    50 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)4307-4320
    Number of pages14
    JournalDiscrete and Continuous Dynamical Systems- Series A
    Volume32
    Issue number12
    DOIs
    Publication statusPublished - 2012 Dec

    Keywords

    • Critical exponent
    • Time-dependent damping
    • Wave equation

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Applied Mathematics
    • Analysis

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