Dynamics of the BBM equation with a distribution force in low regularity spaces

Ming Wang, Anping Liu

DOI: http://dx.doi.org/10.12775/TMNA.2017.058


The Benjamin-Bona-Mahony equation with a distribution force on torus is studied in low regularity spaces. The global well-posedness and the existence of a global attractor in $\dot{H}^{s,p}(\mathbb{T})$ are proved.


Global attractor; Benjamin-Bona-Mahony equation; low regularity

Full Text:



J. Arrieta, A.N. Carvalho and J.K. Hale, A damped hyperbolic equation with critical exponent, Commun. Partial Differential Equations 17 (1992), 841–866.

J.M. Arrieta, J.W. Cholewa, T. Dlotko, A. Rodrı́guez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal. Theor. 56 (2004), 515–554.

J. Avrin, The generalized Benjamin–Bona–Mahony equation in Rn with singular initial data, Nonlinear Anal. 11 (1987), 139–147.

J. Avrin and J.A. Goldstein, Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions, Nonlinear Anal. 9 (1985), 861–865.

T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. B 272 (1972), 47–78.

J.L. Bona and V.A. Dougalis, An initial-and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl. 75 (1980), 503–522.

J.L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst. 23 (2009), 1241–1252.

A.N. Carvalho and J.W. Cholewa, Strongly damped wave equations in W 1,p , Dyn. Syst. (2007), 230–239.

A. Celebi, V.K. Kalantarov and M. Polat, Attractors for the generalized Benjamin–Bona–Mahony equation, J. Differential Equations 157 (1999), 439–451.

Y. Chen, Remark on the global existence for the generalized Benjamin–Bona–Mahony equations in arbitrary dimension, Appl. Anal. 30 (1988), 1–15.

V.V. Chepyzhov, A.A. Ilyin and S.V. Zelik, Strong trajectory and global W 1,p -attractors for the damped-driven Euler system in R2 , Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 1835–1855.

I. Chueshov, M. Polat and S. Siegmund, Gevrey regularity of global attractor for generalized Benjamin–Bona–Mahony equation, Mat. Fiz. Anal. Geom. 11 (2004), 226–242.

J. Colliander, M. Keel, G. Staffilani, H.Takaoka and T. Tao, Sharp global wellposedness for KdV and modified KdV on R and T , J. Amer. Math. Soc. 16 (2003), 705–749.

F. Dell’Oro, Y. Mammeria and V. Pata, The Benjamin–Bona–Mahony equation with dissipative memory, NoDEA Nonlinear Differential Equations Appl. 4 (2015), 899–910.

L. Grafakos, Classical Fourier Analysis, Vol. 2, Springer, New York, 2008.

J. Hale, Asmptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.

V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré 17 (2016), 2555–2584.

C. Liu and F. Meng, Global well-posedness and attractor for damped wave equation with sup-cubic nonlinearity and lower regular forcing on R3 , Topol. Methods Nonlinear Anal. 49 (2017), 551–563.

M. Mei, Large-time behavior of solution for generalized Benjamin–Bona–Mahony–Burgers equations, Nonlinear Anal. 33 (1998), 699–714.

M. Mei, Lq -decay rates of solutions for Benjamin–Bona–Mahony–Burgers equations, J. Differential Equations 158 (1999), 314–340.

M. Stanislavova, On the global attractor for the damped Benjamin–Bona–Mahony equation, Proceedings of the Fifth International Conference on Dynamical Systems and Differential Equations, June 16–19, 2004, Pomona, CA, USA.

M. Stanislavova, A. Stefanov and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin–Bona–Mahony equation on R3 , J. Differential Equations 219 (2005), 451–483.

C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations 227 (2006), 427–443.

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal. Theor. 63 (2005), 49–65.

B. Wang, Strong attractors for the Benjamin–Bona–Mahony equation, Appl. Math. Lett. 10 (1997), 23–28.

B. Wang, Regularity of attractors for the Benjamin–Bona–Mahony equation, J. Phys. A Math. Gen. 31 (1998), 7635–7645.

B. Wang, D. Fussner and C. Bi, Existence of global attractors for the Benjamin–Bona–Mahony equation in unbounded domains, J. Phys. A Math. Theor. 40 (2007), 10491–10504.

B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin–Bona–Mahony equation, J. Phys. A Math. Gen. 30 (1997), 4877–4885.

M. Wang, Long time dynamics for a damped Benjamin–Bona–Mahony equation in low regularity spaces, Nonlinear Anal. Theor. 105 (2014), 134–144.

M. Wang, Long time behavior of a damped generalized BBM equation in low regularity spaces, Math. Method Appl. Sci. 38 (2015), 4852–4866.

M. Wang, Sharp global well-posedness of the BBM equation in Lp type Sobolev spaces, Discrete Contin. Dyn. Syst. Ser. A. 36 (2016), 5763–5788.

M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear Anal. Real World Appl. 11 (2010), 913–919.

Y. Xie, Q. Li, C. Huang et al., Attractors for the semilinear reaction-diffusion equation with distribution derivatives, J. Math. Phys. 54 (2013), 092701.

C. Zhu, Global attractors for the damped Benjamin–Bona–Mahony equation on R1 , Appl. Anal. 86 (2007), 59–61.


  • There are currently no refbacks.

Partnerzy platformy czasopism