American Institute of Mathematical Sciences

Advanced
November  2017, 37(11): 5631-5649. doi: 10.3934/dcds.2017244

The initial-boundary value problems for a class of sixth order nonlinear wave equation

Runzhang Xu 1,2,, , Mingyou Zhang 3, , Shaohua Chen 4, , Yanbing Yang 1, and  Jihong Shen 1,

1. 

College of Science, Harbin Engineering University, Heilongjiang, Harbin 150001, China
2. 

The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China
3. 

College of Automation, Harbin Engineering University, Heilongjiang, Harbin 150001, China
4. 

Department of Mathematics, Cape Breton University, Sydney, NS, B1P 6L2, Canada

* Corresponding author: Runzhang Xu, xurunzh@163.com.

Received  March 2017 Revised  June 2017 Published  July 2017

Fund Project: This work was supported by the National Natural Science Foundation of China (11471087), the China Postdoctoral Science Foundation(2013M540270), the Fundamental Research Funds for the Central Universities.

This paper considers the initial boundary value problem of solutions for a class of sixth order 1-D nonlinear wave equations. We discuss the probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and non-global existence of solutions at three different initial energy levels, i.e., sub-critical level, critical level and sup-critical level.

Keywords: Initial boundary value problem, sixth order wave equation, blow up, existence of global solutions, arbitrarily positive initial energy.
Mathematics Subject Classification: Primary: 35L35, 35A01; Secondary: 35B44, 35L75.
Citation: 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 & Continuous Dynamical Systems, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244
References:
[1]

J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar

[2]

J. V. Boussinesq, Théorie des ondes et des remous qui se propagent le long d$'$un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108.   Google Scholar

[3]

Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Discrete Contin. Dyn. Syst., 17 (2007), 691-711.  doi: 10.3934/dcds.2007.17.691.  Google Scholar

[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.  Google Scholar

[5]

X. J. Han and G. W. Chen, Initial boundary value problem for a class of nonlinear wave equation of higher order, Acta Math. Sci., 27 (2007), 624-640 (in Chinese).   Google Scholar

[6]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.  Google Scholar

[7]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_{tt} = Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[8]

Y. C. Liu and R. Z. Xu, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Appl., 338 (2008), 1169-1187.  doi: 10.1016/j.jmaa.2007.05.076.  Google Scholar

[9]

Y. C. Liu and R. Z. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Physica D., 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.  Google Scholar

[10]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.  Google Scholar

[11]

Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.  doi: 10.1090/S0002-9947-1960-0111898-8.  Google Scholar

[12]

L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[13]

P. Rosenau, Dynamics of dense lattices, Phys. Rev. B. Condensed Matter. Third Series, 36 (1987), 5868-5876.  doi: 10.1103/PhysRevB.36.5868.  Google Scholar

[14]

J. H. Shen, Y. B. Yang and R. Z. Xu, Global existence of solutions for 1-D nonlinear wave equation of sixth order at high initial energy level, Bound. Value Probl., 2014 (2014), 1-6.  doi: 10.1186/1687-2770-2014-31.  Google Scholar

[15]

V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dyn. Syst., 4 (1998), 431-444.  doi: 10.3934/dcds.1998.4.431.  Google Scholar

[16]

V. Varlamov, Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete Contin. Dyn. Syst., 7 (2001), 675-702.  doi: 10.3934/dcds.2001.7.675.  Google Scholar

[17]

H. W. Wang and S. B. Wang, Decay and scattering of small solutions for Rosenau equation, Appl. Math. Comput., 218 (2011), 115-123.  doi: 10.1016/j.amc.2011.05.060.  Google Scholar

[18]

Y. Z. Wang and Y. X. Wang, Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order, Nonlinear Anal., 72 (2010), 4500-4507.  doi: 10.1016/j.na.2010.02.025.  Google Scholar

[19]

S. B. Wang and G. X. Xu, The Cauchy problem for the Rosenau equation, Nonlinear Anal., 71 (2009), 456-466.  doi: 10.1016/j.na.2008.10.085.  Google Scholar

[20]

R. Z. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[21]

R. Z. Xu, Y. C. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal., 71 (2009), 4977-4983.  doi: 10.1016/j.na.2009.03.069.  Google Scholar

[22]

R. Z. Xu, X. C. Wang, H. C. Xu and M. Y. Zhang, Arbitrary energy global existence for wave equation with combined power-type nonlinearities of different signs, Bound. Value Probl., 2016 (2016), 1-6.  doi: 10.1186/s13661-016-0722-4.  Google Scholar

[23]

R. Z. Xu and Y. B. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level, Internat. J. Math., 23 (2012), 1250060, 10 pp. doi: 10.1142/S0129167X12500607.  Google Scholar

show all references

References:
[1]

J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar

[2]

J. V. Boussinesq, Théorie des ondes et des remous qui se propagent le long d$'$un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108.   Google Scholar

[3]

Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Discrete Contin. Dyn. Syst., 17 (2007), 691-711.  doi: 10.3934/dcds.2007.17.691.  Google Scholar

[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.  Google Scholar

[5]

X. J. Han and G. W. Chen, Initial boundary value problem for a class of nonlinear wave equation of higher order, Acta Math. Sci., 27 (2007), 624-640 (in Chinese).   Google Scholar

[6]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.  Google Scholar

[7]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_{tt} = Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[8]

Y. C. Liu and R. Z. Xu, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Appl., 338 (2008), 1169-1187.  doi: 10.1016/j.jmaa.2007.05.076.  Google Scholar

[9]

Y. C. Liu and R. Z. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Physica D., 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.  Google Scholar

[10]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.  Google Scholar

[11]

Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.  doi: 10.1090/S0002-9947-1960-0111898-8.  Google Scholar

[12]

L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[13]

P. Rosenau, Dynamics of dense lattices, Phys. Rev. B. Condensed Matter. Third Series, 36 (1987), 5868-5876.  doi: 10.1103/PhysRevB.36.5868.  Google Scholar

[14]

J. H. Shen, Y. B. Yang and R. Z. Xu, Global existence of solutions for 1-D nonlinear wave equation of sixth order at high initial energy level, Bound. Value Probl., 2014 (2014), 1-6.  doi: 10.1186/1687-2770-2014-31.  Google Scholar

[15]

V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dyn. Syst., 4 (1998), 431-444.  doi: 10.3934/dcds.1998.4.431.  Google Scholar

[16]

V. Varlamov, Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete Contin. Dyn. Syst., 7 (2001), 675-702.  doi: 10.3934/dcds.2001.7.675.  Google Scholar

[17]

H. W. Wang and S. B. Wang, Decay and scattering of small solutions for Rosenau equation, Appl. Math. Comput., 218 (2011), 115-123.  doi: 10.1016/j.amc.2011.05.060.  Google Scholar

[18]

Y. Z. Wang and Y. X. Wang, Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order, Nonlinear Anal., 72 (2010), 4500-4507.  doi: 10.1016/j.na.2010.02.025.  Google Scholar

[19]

S. B. Wang and G. X. Xu, The Cauchy problem for the Rosenau equation, Nonlinear Anal., 71 (2009), 456-466.  doi: 10.1016/j.na.2008.10.085.  Google Scholar

[20]

R. Z. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[21]

R. Z. Xu, Y. C. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal., 71 (2009), 4977-4983.  doi: 10.1016/j.na.2009.03.069.  Google Scholar

[22]

R. Z. Xu, X. C. Wang, H. C. Xu and M. Y. Zhang, Arbitrary energy global existence for wave equation with combined power-type nonlinearities of different signs, Bound. Value Probl., 2016 (2016), 1-6.  doi: 10.1186/s13661-016-0722-4.  Google Scholar

[23]

R. Z. Xu and Y. B. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level, Internat. J. Math., 23 (2012), 1250060, 10 pp. doi: 10.1142/S0129167X12500607.  Google Scholar

Figure 1.   
Figure Options

Download full-size image

Download as PowerPoint slide

[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 & Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072
[2]

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 & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[3]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
[4]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019
[5]

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 & Continuous Dynamical Systems - S, 2020, 13 (7) : 2095-2107. doi: 10.3934/dcdss.2020160
[6]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[7]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[8]

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 & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021107
[9]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117
[10]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431
[11]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
[12]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205
[13]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[14]

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
[15]

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
[16]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[17]

Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032
[18]

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 & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[19]

Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105
[20]

Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (391)
  • HTML views (95)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences