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July  2020, 19(7): 3885-3900. doi: 10.3934/cpaa.2020143

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

Mohamed Jleli and  Bessem Samet ,

Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

* Corresponding author

Received  July 2019 Revised  December 2019 Published  April 2020

Fund Project: B. Samet is supported by Researchers Supporting Project RSP-2019/4, King Saud University, Riyadh, Saudi Arabia

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.
Keywords: Semilinear wave equation, time-dependent damping, exterior domain, blow-up, critical exponent.
Mathematics Subject Classification: Primary: 35B33, 35L71; Secondary: 35L05.
Citation: Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Lin, K. 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.  Google Scholar

[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.   Google Scholar

[8]

K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math., 34 (2011), 327-343.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Lin, K. 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.  Google Scholar

[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.   Google Scholar

[8]

K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math., 34 (2011), 327-343.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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