-
Previous Article
A trace theorem for Sobolev spaces on the Sierpinski gasket
- CPAA Home
- This Issue
-
Next Article
Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source
Blow-up for semilinear wave equations with time-dependent damping in an exterior domain
Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia |
$ \partial_{tt}u-\Delta u +\mu (1+t)^{-\beta} \partial_t u = |u|^p, \quad (t, x)\in (0, \infty)\times D^c, $ |
$ D^c = \mathbb{R}^N\backslash D $ |
$ D $ |
$ \mathbb{R}^N $ |
$ N\geq 2 $ |
$ \mu>0 $ |
$ p>1 $ |
$ -1<\beta<1 $ |
$ 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, $ |
$ n^+ $ |
$ D^c $ |
$ \partial D $ |
$ b $ |
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. |
[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. 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. |
[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. |
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. |
[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. 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. |
[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. |
[1] |
Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 |
[2] |
Ahmad Z. Fino, Mohamed Ali Hamza. Blow-up of solutions to semilinear wave equations with a time-dependent strong damping. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022006 |
[3] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011 |
[4] |
Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 |
[5] |
Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 |
[6] |
Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 |
[7] |
Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 |
[8] |
Donghao Li, Hongwei Zhang, Shuo Liu, Qingiyng Hu. Blow-up of solutions to a viscoelastic wave equation with nonlocal damping. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022009 |
[9] |
Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 |
[10] |
Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 |
[11] |
Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 |
[12] |
Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351 |
[13] |
Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations and Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37 |
[14] |
Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108 |
[15] |
Xudong Luo, Qiaozhen Ma. The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021253 |
[16] |
Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 |
[17] |
Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072 |
[18] |
Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205 |
[19] |
Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069 |
[20] |
Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]