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January  2015, 35(1): 117-137. doi: 10.3934/dcds.2015.35.117

Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation

 1 Laboratoire de Mathematiques Appliquées,UFRMI, Université d'Abidjan Cocody, 22 BP 582 Abidjan 22, Ivory Coast (Cote D'Ivoire)

Received  July 2010 Revised  June 2014 Published  August 2014

We study the existence, the decay and the blow-up of solutions to the Cauchy problem for the multi-dimensional generalized sixth-order Boussinesq equation: $$u_{tt} - \Delta u - \Delta^{2} u- \mu \Delta ^{3} u = \Delta f(u),\; t>0, \; x \in {\mathbb{R}^{n}}, n \geq 1,$$ where $f(u)= \gamma |u|^{p-1}u, \; \gamma \in \mathbb{R}, \; p \geq 2, \; \mu > 1/4$. We find two global existence results for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$ On the other hand we show that if $\mu= 1/3$ and $p>13/2$, then the solution with small initial data decays in time. A blow up in finite time result is also obtained for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$
Citation: Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117
References:
 [1] J. V. Boussinesq, Theorie de l'intermenscence liquide appelee onde solitaire ou de translation, se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris, 72 (1871), 755-799. [2] Y. Cho and T. Ozawa, Remarks on the Modified improved Boussinesq equations in one space dimension, Proceeding of Royal society A, 462 (2006), 1949-1963. doi: 10.1098/rspa.2006.1675. [3] F. M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de-Vries equation, Journ. funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. [4] C. I. Christov, G. A Maugin and M. G. Velande, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev., E54 (1996), 3621-3638. doi: 10.1103/PhysRevE.54.3621. [5] C. I. Christov, G. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation, C. R. Mecanique, 335 (2007), 521-535. doi: 10.1016/j.crme.2007.08.006. [6] P. Daripa and W. Hua, A numerical method for solving an ill posed Boussinesq equation arising in water waves and nonlinear lattice, Appl. Math. Comput., 101 (1999), 159-207. doi: 10.1016/S0096-3003(98)10070-X. [7] P. Daripa and R. K. Dash, Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation, Math. Comput. Simulation, 55 (2001), 393-405. doi: 10.1016/S0378-4754(00)00288-3. [8] P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves, Euro. J. Mech. B Fluids, 25 (2006), 1008-1021. doi: 10.1016/j.euromechflu.2006.02.003. [9] A. Dé Godefroy, Blow up of solutions of a generalized Boussinesq equation, IMA Journ. Math. Appl. Math, 60 (1998), 123-138. doi: 10.1093/imamat/60.2.123. [10] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, 1997. doi: 10.1017/CBO9780511624056. [11] O. Y. Kamenov, Exact periodic solutions of sixth-order generalized Boussinesq equation, J. Phys. A: Math. Theor., 42 (2009), 375501, 11 pp. doi: 10.1088/1751-8113/42/37/375501. [12] T. Kato, On nonlinear Schrodinger equation II. $\mathbbH^s$ solutions and unconditional well posedness, J. Anal. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794. [13] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t}= -Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. [14] Y. Liu, Existence and Blow up of solutions of a nonlinear Pocchahammer-Chree equation, Indianna Univ. Math. Journ., 45 (1996), 797-815. doi: 10.1512/iumj.1996.45.1121. [15] V. G. Makhankov, Dynamics of classical solutions (in non integrable systems), Physics Reports, 35 (1978), 1-128. doi: 10.1016/0370-1573(78)90074-1. [16] G. A. Maugin, Nonlinear Waves in Elastic Crystal, Oxford Mathematical Monographs Series, Oxford University Press, Oxford, 1999. [17] R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. Ser. A, 340, (1992), 47-94. doi: 10.1098/rsta.1992.0055. [18] M. Reed, Abstract Nonlinear Wave Equations, Lecture Notes in Mathematics, 507, Springer-Verlag, Berlin-New York, 1976. [19] G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.

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References:
 [1] J. V. Boussinesq, Theorie de l'intermenscence liquide appelee onde solitaire ou de translation, se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris, 72 (1871), 755-799. [2] Y. Cho and T. Ozawa, Remarks on the Modified improved Boussinesq equations in one space dimension, Proceeding of Royal society A, 462 (2006), 1949-1963. doi: 10.1098/rspa.2006.1675. [3] F. M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de-Vries equation, Journ. funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. [4] C. I. Christov, G. A Maugin and M. G. Velande, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev., E54 (1996), 3621-3638. doi: 10.1103/PhysRevE.54.3621. [5] C. I. Christov, G. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation, C. R. Mecanique, 335 (2007), 521-535. doi: 10.1016/j.crme.2007.08.006. [6] P. Daripa and W. Hua, A numerical method for solving an ill posed Boussinesq equation arising in water waves and nonlinear lattice, Appl. Math. Comput., 101 (1999), 159-207. doi: 10.1016/S0096-3003(98)10070-X. [7] P. Daripa and R. K. Dash, Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation, Math. Comput. Simulation, 55 (2001), 393-405. doi: 10.1016/S0378-4754(00)00288-3. [8] P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves, Euro. J. Mech. B Fluids, 25 (2006), 1008-1021. doi: 10.1016/j.euromechflu.2006.02.003. [9] A. Dé Godefroy, Blow up of solutions of a generalized Boussinesq equation, IMA Journ. Math. Appl. Math, 60 (1998), 123-138. doi: 10.1093/imamat/60.2.123. [10] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, 1997. doi: 10.1017/CBO9780511624056. [11] O. Y. Kamenov, Exact periodic solutions of sixth-order generalized Boussinesq equation, J. Phys. A: Math. Theor., 42 (2009), 375501, 11 pp. doi: 10.1088/1751-8113/42/37/375501. [12] T. Kato, On nonlinear Schrodinger equation II. $\mathbbH^s$ solutions and unconditional well posedness, J. Anal. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794. [13] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t}= -Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. [14] Y. Liu, Existence and Blow up of solutions of a nonlinear Pocchahammer-Chree equation, Indianna Univ. Math. Journ., 45 (1996), 797-815. doi: 10.1512/iumj.1996.45.1121. [15] V. G. Makhankov, Dynamics of classical solutions (in non integrable systems), Physics Reports, 35 (1978), 1-128. doi: 10.1016/0370-1573(78)90074-1. [16] G. A. Maugin, Nonlinear Waves in Elastic Crystal, Oxford Mathematical Monographs Series, Oxford University Press, Oxford, 1999. [17] R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. Ser. A, 340, (1992), 47-94. doi: 10.1098/rsta.1992.0055. [18] M. Reed, Abstract Nonlinear Wave Equations, Lecture Notes in Mathematics, 507, Springer-Verlag, Berlin-New York, 1976. [19] G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.
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