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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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

E. Ott and R. N. Sudan, Damping of solitary waves,, The Physics of fluids, 13 (1970). doi: 10.1063/1.1693097. Google Scholar

[15]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[16]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcial. Ekvac., 54 (2011), 119. doi: 10.1619/fesi.54.119. Google Scholar

[17]

S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations,, Asymptot. Anal., 68 (2010), 155. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

E. Ott and R. N. Sudan, Damping of solitary waves,, The Physics of fluids, 13 (1970). doi: 10.1063/1.1693097. Google Scholar

[15]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[16]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcial. Ekvac., 54 (2011), 119. doi: 10.1619/fesi.54.119. Google Scholar

[17]

S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations,, Asymptot. Anal., 68 (2010), 155. Google Scholar

[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. Google Scholar

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