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The index bundle and multiparameter bifurcation for discrete dynamical systems
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 |
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.
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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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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