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

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.

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 and 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.

[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. 

[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.

[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]

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). 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

J. H. ShenY. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

R. Z. XuY. 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.

[22]

R. Z. XuX. C. WangH. 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.

[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.

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.

[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. 

[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.

[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]

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). 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

J. H. ShenY. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

R. Z. XuY. 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.

[22]

R. Z. XuX. C. WangH. 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.

[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.

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