# American Institute of Mathematical Sciences

October  2021, 26(10): 5321-5335. doi: 10.3934/dcdsb.2020345

## Singular support of the global attractor for a damped BBM equation

 a. College of Science, National University of Defense Technology, Changsha, 410073, China b. TAG_SUPSchool of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China c. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

* Corresponding author: Ming Wang

Received  June 2020 Published  October 2021 Early access  November 2020

Fund Project: The research was supported in part by the National Natural Science Foundations of China (Grant Nos. 11701535, 11771449 and 11471129), China Postdoctoral Science Foundation No. 2019T120966, the Fundamental Research Funds for the Central Universities, China University of Geosciences(Wuhan)(No. CUGSX01), and the Natural Science Foundation of Hunan Province No. 2020JJ4102

The singular support of the global attractor is introduced. It is shown that the singular support of the global attractor for a damped BBM equation equals to the singular support of the force term. This gives a delicate description of the local regularity, which roughly says that the attractor is smooth exactly where the force is smooth.

Citation: Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
##### References:
 [1] K. Ammari and E. Crépeau, Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 127 (2019), 39-43.  doi: 10.1016/j.sysconle.2019.03.005. [2] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [3] 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.  doi: 10.1016/0022-247X(80)90098-0. [4] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [5] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst, 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241. [6] F. Dell'Oro, O. Goubet, Y. Mammeri and V. Pata, Global attractors for the Benjamin-Bona-Mahony equation with memory, Indiana Univ. Math. J., 69 (2020), 749-783.  doi: 10.1512/iumj.2020.69.7906. [7] F. Dell'Oro and Y. Mammeri, Benjamin-Bona-Mahony equations with memory and rayleigh friction, Applied Mathematics & Optimization, (2019), in press. doi: 10.1007/s00245-019-09568-z. [8] F. Dell'Oro, Y. Mammeri and V. Pata, The Benjamin-Bona-Mahony equation with dissipative memory, Nonlinear Differential Equations and Applications NoDEA, 22 (2015), 899-910.  doi: 10.1007/s00030-014-0308-8. [9] L. C. Evans, Partial Differential Equations, American Mathematical Soc., Providence, RI, 2010. doi: 10.1090/GSM/019. [10] O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644.  doi: 10.3934/dcds.2000.6.625. [11] O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, Journal of Differential Equations, 185 (2002), 25–53. doi: 10.1006/jdeq.2001.4163. [12] Y. Guo, M. Wang and Y. Tang, Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R, Applicable Analysis: An International Journal, 94 (2015), 1766-1783.  doi: 10.1080/00036811.2014.946561. [13] J. K. Hale, Asmptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. [14] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, 1990. doi: 10.1007/978-3-642-61497-2. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2005. doi: 10.1007/b138375. [16] J.-R. Kang, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Applied Mathematics and Computation, 274 (2016), 343–352. doi: 10.1016/j.amc.2015.10.086. [17] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, Journal of the American Mathematical Society, 4 (1991) 323–347. doi: 10.1090/S0894-0347-1991-1086966-0. [18] D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23–100. doi: 10.4171/rmi/1049. [19] Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stochastics and Dynamics, 18 (2018), 1850004. doi: 10.1142/S0219493718500041. [20] Y. Qin, X. Yang and X. Liu, Pullback attractor of Benjamin-Bona-Mahony equations in $H^2$, Acta. Math. Sci., 32 (2012), 1338-1348.  doi: 10.1016/S0252-9602(12)60103-9. [21] M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. suppl., 2005 (2005), 824-832. [22] M. Stanislavova, A. Stefanov and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on ${\mathbb{R}}^3$, J. Differ. Equations, 219 (2005), 451-483.  doi: 10.1016/j.jde.2005.08.004. [23] C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differ. Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010. [24] 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. [25] B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 31 (1998), 7635-7645.  doi: 10.1088/0305-4470/31/37/021. [26] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, Journal of Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012. [27] B. Wang, D. W. 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.  doi: 10.1088/1751-8113/40/34/007. [28] B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 30 (1997), 4877-4885.  doi: 10.1088/0305-4470/30/13/035. [29] M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Analysis: Theory, Methods & Applications, 105 (2014), 134-144.  doi: 10.1016/j.na.2014.04.013. [30] M. Wang, Long time behavior of a damped generalized BBM equation in low regularity spaces, Math. Method App. Sci., 38 (2015), 4852-4866.  doi: 10.1002/mma.3400. [31] M. Wang, Global attractor for weakly damped gKdV equations in higher Sobolev spaces, Discrete Contin. Dyn. Syst.-A., 35 (2015), 3799-3825.  doi: 10.3934/dcds.2015.35.3799. [32] M. Wang, Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces, Discrete Cont. Dyn-A., 36 (2016), 5763-5788.  doi: 10.3934/dcds.2016053. [33] M. Wang and A. Liu, Dynamics of the BBM equation with a distribution force in low regularity spaces, Topological Methods in Nonlinear Analysis, 51 (2018), 91-109.  doi: 10.12775/TMNA.2017.058. [34] M. Wang and Z. Zhang, Sharp global well-posedness for the fractional BBM equation, Mathematical Methods in the Applied Sciences, 41 (2018), 5906-5918.  doi: 10.1002/mma.5109. [35] Q. Zhang and Y. Li, Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, Journal of Dynamical and Control Systems, 26 (2020), 423-441.  doi: 10.1007/s10883-019-09450-9. [36] M. Zhao, X.-G. Yang, X. Yan and X. Cui, Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operator, Asymptotic Analysis, (2020), in press. doi: 10.3233/ASY-201601.

show all references

##### References:
 [1] K. Ammari and E. Crépeau, Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 127 (2019), 39-43.  doi: 10.1016/j.sysconle.2019.03.005. [2] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [3] 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.  doi: 10.1016/0022-247X(80)90098-0. [4] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [5] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst, 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241. [6] F. Dell'Oro, O. Goubet, Y. Mammeri and V. Pata, Global attractors for the Benjamin-Bona-Mahony equation with memory, Indiana Univ. Math. J., 69 (2020), 749-783.  doi: 10.1512/iumj.2020.69.7906. [7] F. Dell'Oro and Y. Mammeri, Benjamin-Bona-Mahony equations with memory and rayleigh friction, Applied Mathematics & Optimization, (2019), in press. doi: 10.1007/s00245-019-09568-z. [8] F. Dell'Oro, Y. Mammeri and V. Pata, The Benjamin-Bona-Mahony equation with dissipative memory, Nonlinear Differential Equations and Applications NoDEA, 22 (2015), 899-910.  doi: 10.1007/s00030-014-0308-8. [9] L. C. Evans, Partial Differential Equations, American Mathematical Soc., Providence, RI, 2010. doi: 10.1090/GSM/019. [10] O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644.  doi: 10.3934/dcds.2000.6.625. [11] O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, Journal of Differential Equations, 185 (2002), 25–53. doi: 10.1006/jdeq.2001.4163. [12] Y. Guo, M. Wang and Y. Tang, Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R, Applicable Analysis: An International Journal, 94 (2015), 1766-1783.  doi: 10.1080/00036811.2014.946561. [13] J. K. Hale, Asmptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. [14] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, 1990. doi: 10.1007/978-3-642-61497-2. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2005. doi: 10.1007/b138375. [16] J.-R. Kang, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Applied Mathematics and Computation, 274 (2016), 343–352. doi: 10.1016/j.amc.2015.10.086. [17] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, Journal of the American Mathematical Society, 4 (1991) 323–347. doi: 10.1090/S0894-0347-1991-1086966-0. [18] D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23–100. doi: 10.4171/rmi/1049. [19] Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stochastics and Dynamics, 18 (2018), 1850004. doi: 10.1142/S0219493718500041. [20] Y. Qin, X. Yang and X. Liu, Pullback attractor of Benjamin-Bona-Mahony equations in $H^2$, Acta. Math. Sci., 32 (2012), 1338-1348.  doi: 10.1016/S0252-9602(12)60103-9. [21] M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. suppl., 2005 (2005), 824-832. [22] M. Stanislavova, A. Stefanov and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on ${\mathbb{R}}^3$, J. Differ. Equations, 219 (2005), 451-483.  doi: 10.1016/j.jde.2005.08.004. [23] C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differ. Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010. [24] 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. [25] B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 31 (1998), 7635-7645.  doi: 10.1088/0305-4470/31/37/021. [26] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, Journal of Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012. [27] B. Wang, D. W. 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.  doi: 10.1088/1751-8113/40/34/007. [28] B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A Math. Gen., 30 (1997), 4877-4885.  doi: 10.1088/0305-4470/30/13/035. [29] M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Analysis: Theory, Methods & Applications, 105 (2014), 134-144.  doi: 10.1016/j.na.2014.04.013. [30] M. Wang, Long time behavior of a damped generalized BBM equation in low regularity spaces, Math. Method App. Sci., 38 (2015), 4852-4866.  doi: 10.1002/mma.3400. [31] M. Wang, Global attractor for weakly damped gKdV equations in higher Sobolev spaces, Discrete Contin. Dyn. Syst.-A., 35 (2015), 3799-3825.  doi: 10.3934/dcds.2015.35.3799. [32] M. Wang, Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces, Discrete Cont. Dyn-A., 36 (2016), 5763-5788.  doi: 10.3934/dcds.2016053. [33] M. Wang and A. Liu, Dynamics of the BBM equation with a distribution force in low regularity spaces, Topological Methods in Nonlinear Analysis, 51 (2018), 91-109.  doi: 10.12775/TMNA.2017.058. [34] M. Wang and Z. Zhang, Sharp global well-posedness for the fractional BBM equation, Mathematical Methods in the Applied Sciences, 41 (2018), 5906-5918.  doi: 10.1002/mma.5109. [35] Q. Zhang and Y. Li, Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, Journal of Dynamical and Control Systems, 26 (2020), 423-441.  doi: 10.1007/s10883-019-09450-9. [36] M. Zhao, X.-G. Yang, X. Yan and X. Cui, Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operator, Asymptotic Analysis, (2020), in press. doi: 10.3233/ASY-201601.
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