Long-time behavior of solutions of a BBM equation with generalized damping

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September 2015, 20(7): 1897-1915. doi: 10.3934/dcdsb.2015.20.1897

Long-time behavior of solutions of a BBM equation with generalized damping

Jean-Paul Chehab ^1, , Pierre Garnier ^2, and Youcef Mammeri ^2,

1.	LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens
2.	Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France, France

Received February 2014 Revised May 2015 Published July 2015

Abstract
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We study the long-time behavior of the solution of a damped BBM equation $u_t + u_x - u_{xxt} + uu_x + \mathscr{L}_{\gamma}(u) = 0$. The proposed dampings $\mathscr{L}_{\gamma}$ generalize standards ones, as parabolic ($\mathscr{L}_{\gamma}(u)=-\Delta u$) or weak damping ($\mathscr{L}_{\gamma}(u)=\gamma u$) and allows us to consider a greater range. After establish the local well-posedness in the energy space, we investigate some numerical properties.

Keywords: dissipation, damping, dispersion, BBM equation, local well-posedness..

Mathematics Subject Classification: 35B40, 35Q53, 76B03, 76B1.

Citation: Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897

References:

[1]	M. Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems,, Commun. Pure Appl. Anal., 7 (2008), 211.
[2]	C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations,, J. Differential Equations, 81 (1989), 1. doi: 10.1016/0022-0396(89)90176-9.
[3]	T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47. doi: 10.1098/rsta.1972.0032.
[4]	M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265. doi: 10.1016/j.physd.2004.01.023.
[5]	J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, Commun. Pure Appl. Anal., 12 (2013), 519. doi: 10.3934/cpaa.2013.12.519.
[6]	J.-P. Chehab and G. Sadaka, On damping rates of dissipative KdV equations,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487. doi: 10.3934/dcdss.2013.6.1487.
[7]	A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation,, IMA J. Numer. Anal., 20 (2000), 235. doi: 10.1093/imanum/20.2.235.
[8]	J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,, J. Differential Equations, 74 (1988), 369. doi: 10.1016/0022-0396(88)90010-1.
[9]	J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations,, J. Differential Equations, 110 (1994), 356. doi: 10.1006/jdeq.1994.1071.
[10]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.
[11]	O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25. doi: 10.1006/jdeq.2001.4163.
[12]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, Large time asymptotics for the BBM-Burgers equation,, Ann. Henri Poincaré, 8 (2007), 485. doi: 10.1007/s00023-006-0314-4.
[13]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationnary waves,, Phil. Maj., 39 (1895), 422.
[14]	E. Ott and R. N. Sudan, Damping of solitary waves,, The Physics of fluids, 13 (1970). doi: 10.1063/1.1693097.
[15]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3.
[16]	S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcial. Ekvac., 54 (2011), 119. doi: 10.1619/fesi.54.119.
[17]	S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations,, Asymptot. Anal., 68 (2010), 155.
[18]	B. Wang, Strong attractors for the Benjamin-Bona-Mahony equation,, Appl. Math. Lett., 10 (1997), 23. doi: 10.1016/S0893-9659(97)00005-0.

show all references

References:

[1]	M. Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems,, Commun. Pure Appl. Anal., 7 (2008), 211.
[2]	C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations,, J. Differential Equations, 81 (1989), 1. doi: 10.1016/0022-0396(89)90176-9.
[3]	T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47. doi: 10.1098/rsta.1972.0032.
[4]	M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265. doi: 10.1016/j.physd.2004.01.023.
[5]	J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, Commun. Pure Appl. Anal., 12 (2013), 519. doi: 10.3934/cpaa.2013.12.519.
[6]	J.-P. Chehab and G. Sadaka, On damping rates of dissipative KdV equations,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487. doi: 10.3934/dcdss.2013.6.1487.
[7]	A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation,, IMA J. Numer. Anal., 20 (2000), 235. doi: 10.1093/imanum/20.2.235.
[8]	J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,, J. Differential Equations, 74 (1988), 369. doi: 10.1016/0022-0396(88)90010-1.
[9]	J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations,, J. Differential Equations, 110 (1994), 356. doi: 10.1006/jdeq.1994.1071.
[10]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.
[11]	O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25. doi: 10.1006/jdeq.2001.4163.
[12]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, Large time asymptotics for the BBM-Burgers equation,, Ann. Henri Poincaré, 8 (2007), 485. doi: 10.1007/s00023-006-0314-4.
[13]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationnary waves,, Phil. Maj., 39 (1895), 422.
[14]	E. Ott and R. N. Sudan, Damping of solitary waves,, The Physics of fluids, 13 (1970). doi: 10.1063/1.1693097.
[15]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3.
[16]	S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcial. Ekvac., 54 (2011), 119. doi: 10.1619/fesi.54.119.
[17]	S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations,, Asymptot. Anal., 68 (2010), 155.
[18]	B. Wang, Strong attractors for the Benjamin-Bona-Mahony equation,, Appl. Math. Lett., 10 (1997), 23. doi: 10.1016/S0893-9659(97)00005-0.

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