Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 35B40, 35Q53, 76B03, 76B15.


    \begin{equation} \\ \end{equation}
  • [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), 211-227.


    C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.doi: 10.1016/0022-0396(89)90176-9.


    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), 47-78.doi: 10.1098/rsta.1972.0032.


    M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.doi: 10.1016/j.physd.2004.01.023.


    J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, Commun. Pure Appl. Anal., 12 (2013), 519-546.doi: 10.3934/cpaa.2013.12.519.


    J.-P. Chehab and G. Sadaka, On damping rates of dissipative KdV equations, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487-1506.doi: 10.3934/dcdss.2013.6.1487.


    A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation, IMA J. Numer. Anal., 20 (2000), 235-261.doi: 10.1093/imanum/20.2.235.


    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), 369-390.doi: 10.1016/0022-0396(88)90010-1.


    J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differential Equations, 110 (1994), 356-359.doi: 10.1006/jdeq.1994.1071.


    O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.


    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), 25-53.doi: 10.1006/jdeq.2001.4163.


    N. Hayashi, E. I. Kaikina and P. I. Naumkin, Large time asymptotics for the BBM-Burgers equation, Ann. Henri Poincaré, 8 (2007), 485-511.doi: 10.1007/s00023-006-0314-4.


    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), 422-443.


    E. Ott and R. N. Sudan, Damping of solitary waves, The Physics of fluids, 13 (1970), p1432.doi: 10.1063/1.1693097.


    R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Applied Mathematical Sciences, 68, Springer, New York, 1997.doi: 10.1007/978-1-4612-0645-3.


    S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcial. Ekvac., 54 (2011), 119-138.doi: 10.1619/fesi.54.119.


    S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations, Asymptot. Anal., 68 (2010), 155-186.


    B. Wang, Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett., 10 (1997), 23-28.doi: 10.1016/S0893-9659(97)00005-0.

  • 加载中

Article Metrics

HTML views() PDF downloads(88) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint