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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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. doi: 10.1098/rspa.2006.1675. Google Scholar

[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. doi: 10.1016/0022-1236(91)90103-C. Google Scholar

[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. doi: 10.1103/PhysRevE.54.3621. Google Scholar

[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. doi: 10.1016/j.crme.2007.08.006. Google Scholar

[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. doi: 10.1016/S0096-3003(98)10070-X. Google Scholar

[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. doi: 10.1016/S0378-4754(00)00288-3. Google Scholar

[8]

P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves,, Euro. J. Mech. B Fluids, 25 (2006), 1008. doi: 10.1016/j.euromechflu.2006.02.003. Google Scholar

[9]

A. Dé Godefroy, Blow up of solutions of a generalized Boussinesq equation,, IMA Journ. Math. Appl. Math, 60 (1998), 123. doi: 10.1093/imamat/60.2.123. Google Scholar

[10]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997). doi: 10.1017/CBO9780511624056. Google Scholar

[11]

O. Y. Kamenov, Exact periodic solutions of sixth-order generalized Boussinesq equation,, J. Phys. A: Math. Theor., 42 (2009). doi: 10.1088/1751-8113/42/37/375501. Google Scholar

[12]

T. Kato, On nonlinear Schrodinger equation II. $\mathbbH^s$ solutions and unconditional well posedness,, J. Anal. Math., 67 (1995), 281. doi: 10.1007/BF02787794. Google Scholar

[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. Google Scholar

[14]

Y. Liu, Existence and Blow up of solutions of a nonlinear Pocchahammer-Chree equation,, Indianna Univ. Math. Journ., 45 (1996), 797. doi: 10.1512/iumj.1996.45.1121. Google Scholar

[15]

V. G. Makhankov, Dynamics of classical solutions (in non integrable systems),, Physics Reports, 35 (1978), 1. doi: 10.1016/0370-1573(78)90074-1. Google Scholar

[16]

G. A. Maugin, Nonlinear Waves in Elastic Crystal,, Oxford Mathematical Monographs Series, (1999). Google Scholar

[17]

R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves,, Phil. Trans. R. Soc. Lond. Ser. A, 340 (1992), 47. doi: 10.1098/rsta.1992.0055. Google Scholar

[18]

M. Reed, Abstract Nonlinear Wave Equations,, Lecture Notes in Mathematics, (1976). Google Scholar

[19]

G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974). Google Scholar

show all references

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. Google Scholar

[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. doi: 10.1098/rspa.2006.1675. Google Scholar

[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. doi: 10.1016/0022-1236(91)90103-C. Google Scholar

[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. doi: 10.1103/PhysRevE.54.3621. Google Scholar

[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. doi: 10.1016/j.crme.2007.08.006. Google Scholar

[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. doi: 10.1016/S0096-3003(98)10070-X. Google Scholar

[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. doi: 10.1016/S0378-4754(00)00288-3. Google Scholar

[8]

P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves,, Euro. J. Mech. B Fluids, 25 (2006), 1008. doi: 10.1016/j.euromechflu.2006.02.003. Google Scholar

[9]

A. Dé Godefroy, Blow up of solutions of a generalized Boussinesq equation,, IMA Journ. Math. Appl. Math, 60 (1998), 123. doi: 10.1093/imamat/60.2.123. Google Scholar

[10]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997). doi: 10.1017/CBO9780511624056. Google Scholar

[11]

O. Y. Kamenov, Exact periodic solutions of sixth-order generalized Boussinesq equation,, J. Phys. A: Math. Theor., 42 (2009). doi: 10.1088/1751-8113/42/37/375501. Google Scholar

[12]

T. Kato, On nonlinear Schrodinger equation II. $\mathbbH^s$ solutions and unconditional well posedness,, J. Anal. Math., 67 (1995), 281. doi: 10.1007/BF02787794. Google Scholar

[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. Google Scholar

[14]

Y. Liu, Existence and Blow up of solutions of a nonlinear Pocchahammer-Chree equation,, Indianna Univ. Math. Journ., 45 (1996), 797. doi: 10.1512/iumj.1996.45.1121. Google Scholar

[15]

V. G. Makhankov, Dynamics of classical solutions (in non integrable systems),, Physics Reports, 35 (1978), 1. doi: 10.1016/0370-1573(78)90074-1. Google Scholar

[16]

G. A. Maugin, Nonlinear Waves in Elastic Crystal,, Oxford Mathematical Monographs Series, (1999). Google Scholar

[17]

R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves,, Phil. Trans. R. Soc. Lond. Ser. A, 340 (1992), 47. doi: 10.1098/rsta.1992.0055. Google Scholar

[18]

M. Reed, Abstract Nonlinear Wave Equations,, Lecture Notes in Mathematics, (1976). Google Scholar

[19]

G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974). Google Scholar

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