Advanced Search
Article Contents
Article Contents

Blow-up for semilinear wave equations with time-dependent damping in an exterior domain

  • * Corresponding author

    * Corresponding author
B. Samet is supported by Researchers Supporting Project RSP-2019/4, King Saud University, Riyadh, Saudi Arabia
Abstract Full Text(HTML) Related Papers Cited by
  • We consider the semilinear wave equation with time-dependent damping

    $ \partial_{tt}u-\Delta u +\mu (1+t)^{-\beta} \partial_t u = |u|^p, \quad (t, x)\in (0, \infty)\times D^c, $

    where $ D^c = \mathbb{R}^N\backslash D $, $ D $ is the closed unit ball in $ \mathbb{R}^N $, $ N\geq 2 $, $ \mu>0 $, $ p>1 $ and $ -1<\beta<1 $. The considered equation is investigated under the boundary conditions:

    $ u(t, x) \left(\mbox{or } \frac{\partial u}{\partial n^+}(t, x)\right) = b(t)f(x)\, \, \mbox{on}\, \, (0, \infty)\times \partial D, $

    where $ n^+ $ is the outward (relative to $ D^c $) unit normal on $ \partial D $. General blow-up results are established for the considered problems. Moreover, for a certain class of functions $ b $, the critical exponent in the sense of Fujita is obtained.

    Mathematics Subject Classification: Primary: 35B33, 35L71; Secondary: 35L05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t= \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo., 13 (1966), 109-124.  doi: 10.15083/00039873.
    [2] R. Ikehata, Global existence of solutions for semilinear damped wave equation in 2-D exterior domain, J. Differ. Equ., 200 (2004), 53-68.  doi: 10.1016/j.jde.2003.08.009.
    [3] R. Ikehata, Two dimensional exterior mixed problem for semilinear damped wave equations, J. Math. Anal. Appl., 301 (2005), 366-377.  doi: 10.1016/j.jmaa.2004.07.028.
    [4] M. Jleli and B. Samet, New blow-up results for nonlinear boundary value problems in exterior domains, Nonlinear Anal., 178 (2019), 348-365.  doi: 10.1016/j.na.2018.09.003.
    [5] N. Laia and S. Yin, Finite time blow-up for a kind of initial-boundary value problem of semilinear damped wave equation, Math. Meth. Appl. Sci., 40 (2017), 1223-1230.  doi: 10.1002/mma.4046.
    [6] J. LinK. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst., 32 (2012), 4307-4320.  doi: 10.3934/dcds.2012.32.4307.
    [7] E. Mitidieri and S.I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362. 
    [8] K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math., 34 (2011), 327-343. 
    [9] T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal., 70 (2009), 3696-3701.  doi: 10.1016/j.na.2008.07.025.
    [10] K. Ono, Decay estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 286 (2003), 540-562.  doi: 10.1016/S0022-247X(03)00489-X.
    [11] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differ. Equ., 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.
    [12] Q. S. Zhang, A general blow-up result on nonlinear boundary-value problems on exterior domains, Proc. R. Soc. Edinb. Sect. A, 131 (2001), 451-475.  doi: 10.1017/S0308210500000950.
    [13] Q. 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.  doi: 10.1016/S0764-4442(01)01999-1.
  • 加载中

Article Metrics

HTML views(655) PDF downloads(272) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint