American Institute of Mathematical Sciences

Advanced
  • 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
May  2019, 18(3): 1351-1358. doi: 10.3934/cpaa.2019065

Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

Yanbing Yang 1,2,3, and  Runzhang Xu 1,,

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

* Corresponding author

Received  June 2018 Revised  September 2018 Published  November 2018

Fund Project: The first author was supported by the National Natural Science Foundation of China (11801114), the Heilongjiang Postdoctoral Foundation (LBH-Z15036), the China Scholarship Council (201706685064), the Fundamental Research Funds for the Central Universities. The second author was supported by the National Natural Science Foundation of China (11871017), the China Postdoctoral Science Foundation (2013M540270), the Fundamental Research Funds for the Central Universities.

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.

Keywords: Wave equation, initial boundary value problem, strong damping term, finite time blow up, supercritical initial energy.
Mathematics Subject Classification: Primary: 35L05, 35L20, 35B44.
Citation: Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
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.

Table 1.  Obtained results and open problems for problem (1)-(3)
Global existence Asymptotic behavior Blow up
Subcritical initial energy $E(0)<d$ Reference [4] Reference [4] Reference [4]
Critical initial energy $E(0)=d$ Reference [4] Reference [4] Reference [4]
Supercritical initial energy $E(0)>d$ Open problem Open problem $\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
Reference [4]		 $\omega>0$, $\mu>0$
$\omega>0$, $\mu=0$
$\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
Present paper
Table Options

Download as excel

Global existence Asymptotic behavior Blow up
Subcritical initial energy $E(0)<d$ Reference [4] Reference [4] Reference [4]
Critical initial energy $E(0)=d$ Reference [4] Reference [4] Reference [4]
Supercritical initial energy $E(0)>d$ Open problem Open problem $\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
Reference [4]		 $\omega>0$, $\mu>0$
$\omega>0$, $\mu=0$
$\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
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

  • PDF downloads (586)
  • HTML views (375)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy