-
Previous Article
A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach
- CPAA Home
- This Issue
-
Next Article
Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations
Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up
1. | College of Science, Harbin Engineering University, 150001, China |
2. | College of Computer Science and Technology, Harbin Engineering University, 150001, China |
3. | Department of Mathematics, University of Texas, Arlington, TX 76019, USA |
By introducing a new increasing auxiliary function and employing the adapted concavity method, this paper presents a finite time blow up result of the solution for the initial boundary value problem of a class of nonlinear wave equations with both strongly and weakly damped terms at supercritical initial energy level.
References:
[1] |
B. Bilgin and V. Kalantarov,
Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94.
doi: 10.1016/j.jmaa.2013.01.056. |
[2] |
L. Bociu and I. Lasiecka,
Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.
doi: 10.1016/j.jde.2010.03.009. |
[3] |
H. Chen and G. Liu,
Well-posedness for a class of Kirchhoff equations with damping and memory terms, IMA J. Appl. Math., 80 (2015), 1808-1836.
doi: 10.1093/imamat/hxv018. |
[4] |
F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.
doi: 10.1016/j.anihpc.2005.02.007. |
[5] |
S. Gerbi and B. Said-Houari,
Exponential decay for solutions to semilinear damped wave equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 559-566.
doi: 10.3934/dcdss.2012.5.559. |
[6] |
S. Gerbi and B. Said-Houari,
Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.
|
[7] |
P. Graber and B. Said-Houari,
Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122.
doi: 10.1007/s00245-012-9165-1. |
[8] |
G. Liu and S. Xia,
Global existence and finite time blow up for a class of semilinear wave equations on R-N, Comput. Math. Appl., 70 (2015), 1345-1356.
doi: 10.1016/j.camwa.2015.07.021. |
[9] |
Y. Liu and R. Xu,
A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.
doi: 10.1016/j.jde.2007.10.015. |
[10] |
G. Philippin,
Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., 66 (2015), 129-134.
doi: 10.1007/s00033-014-0400-2. |
[11] |
B. Said-Houari,
Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (2010), 79-92.
|
[12] |
J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043.
doi: 10.1142/S0129167X13500432. |
[13] |
H. Song and D. Xue,
Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 19 (2014), 245-251.
doi: 10.1016/j.na.2014.06.012. |
[14] |
H. Song and C. Zhong,
Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.
doi: 10.1016/j.nonrwa.2010.02.015. |
[15] |
Y. Wang,
A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.
doi: 10.1090/S0002-9939-08-09514-2. |
[16] |
Y. Wang,
Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA J. Appl. Math., 74 (2009), 392-415.
doi: 10.1093/imamat/hxp004. |
[17] |
T. Wick,
Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity, Comput. Mech., 53 (2014), 29-43.
doi: 10.1007/s00466-013-0890-3. |
[18] |
R. Xu,
Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.
|
[19] |
R. Xu and Y. Yang,
Global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations, Quart. Appl. Math., 71 (2013), 401-415.
doi: 10.1090/s0033-569x-2012-01295-6. |
[20] |
R. Xu, Y. Yang and Y. Liu,
Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.
doi: 10.1080/00036811.2011.601456. |
[21] |
R. Xu, Y. Yang, B. Liu, J. Shen and S. Huang,
Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015), 955-976.
doi: 10.1007/s00033-014-0459-9. |
[22] |
X. Zhu, F. Li and T. Rong,
Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun. Pure Appl. Anal., 14 (2015), 2465-2485.
doi: 10.3934/cpaa.2015.14.2465. |
show all references
References:
[1] |
B. Bilgin and V. Kalantarov,
Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94.
doi: 10.1016/j.jmaa.2013.01.056. |
[2] |
L. Bociu and I. Lasiecka,
Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.
doi: 10.1016/j.jde.2010.03.009. |
[3] |
H. Chen and G. Liu,
Well-posedness for a class of Kirchhoff equations with damping and memory terms, IMA J. Appl. Math., 80 (2015), 1808-1836.
doi: 10.1093/imamat/hxv018. |
[4] |
F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.
doi: 10.1016/j.anihpc.2005.02.007. |
[5] |
S. Gerbi and B. Said-Houari,
Exponential decay for solutions to semilinear damped wave equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 559-566.
doi: 10.3934/dcdss.2012.5.559. |
[6] |
S. Gerbi and B. Said-Houari,
Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.
|
[7] |
P. Graber and B. Said-Houari,
Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122.
doi: 10.1007/s00245-012-9165-1. |
[8] |
G. Liu and S. Xia,
Global existence and finite time blow up for a class of semilinear wave equations on R-N, Comput. Math. Appl., 70 (2015), 1345-1356.
doi: 10.1016/j.camwa.2015.07.021. |
[9] |
Y. Liu and R. Xu,
A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.
doi: 10.1016/j.jde.2007.10.015. |
[10] |
G. Philippin,
Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., 66 (2015), 129-134.
doi: 10.1007/s00033-014-0400-2. |
[11] |
B. Said-Houari,
Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (2010), 79-92.
|
[12] |
J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043.
doi: 10.1142/S0129167X13500432. |
[13] |
H. Song and D. Xue,
Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 19 (2014), 245-251.
doi: 10.1016/j.na.2014.06.012. |
[14] |
H. Song and C. Zhong,
Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.
doi: 10.1016/j.nonrwa.2010.02.015. |
[15] |
Y. Wang,
A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.
doi: 10.1090/S0002-9939-08-09514-2. |
[16] |
Y. Wang,
Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA J. Appl. Math., 74 (2009), 392-415.
doi: 10.1093/imamat/hxp004. |
[17] |
T. Wick,
Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity, Comput. Mech., 53 (2014), 29-43.
doi: 10.1007/s00466-013-0890-3. |
[18] |
R. Xu,
Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.
|
[19] |
R. Xu and Y. Yang,
Global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations, Quart. Appl. Math., 71 (2013), 401-415.
doi: 10.1090/s0033-569x-2012-01295-6. |
[20] |
R. Xu, Y. Yang and Y. Liu,
Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.
doi: 10.1080/00036811.2011.601456. |
[21] |
R. Xu, Y. Yang, B. Liu, J. Shen and S. Huang,
Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015), 955-976.
doi: 10.1007/s00033-014-0459-9. |
[22] |
X. Zhu, F. Li and T. Rong,
Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun. Pure Appl. Anal., 14 (2015), 2465-2485.
doi: 10.3934/cpaa.2015.14.2465. |
Global existence | Asymptotic behavior | Blow up | ||
Subcritical initial energy |
Reference [4] | Reference [4] | Reference [4] | |
Critical initial energy |
Reference [4] | Reference [4] | Reference [4] | |
Supercritical initial energy |
Open problem | Open problem | Reference [4] |
Present paper |
Global existence | Asymptotic behavior | Blow up | ||
Subcritical initial energy |
Reference [4] | Reference [4] | Reference [4] | |
Critical initial energy |
Reference [4] | Reference [4] | Reference [4] | |
Supercritical initial energy |
Open problem | Open problem | Reference [4] |
Present paper |
[1] |
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 |
[2] |
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 |
[3] |
Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 635-648. doi: 10.3934/eect.2021019 |
[4] |
Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2095-2107. doi: 10.3934/dcdss.2020160 |
[5] |
Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086 |
[6] |
Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 |
[7] |
Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 |
[8] |
Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure and Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161 |
[9] |
Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 |
[10] |
Salim A. Messaoudi, Ala A. Talahmeh. Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1233-1245. doi: 10.3934/dcdss.2021107 |
[11] |
Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 |
[12] |
Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005 |
[13] |
Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 |
[14] |
Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319 |
[15] |
Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 |
[16] |
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 |
[17] |
Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917 |
[18] |
Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2621-2634. doi: 10.3934/dcdsb.2021151 |
[19] |
Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130 |
[20] |
Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]