Singular support of the global attractor for a damped BBM equation

Jianhua Huang; Yanbin Tang; Ming Wang

doi:10.3934/dcdsb.2020345

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

Singular support of the global attractor for a damped BBM equation

Jianhua Huang ^a, , Yanbin Tang ^b, and Ming Wang ^c,a,,

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 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.

Keywords: BBM equation, global attractor, singular support, local regularity, unbounded domain.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 37L30.

Citation: Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. C. Evans, Partial Differential Equations, American Mathematical Soc., Providence, RI, 2010. doi: 10.1090/GSM/019. Google Scholar
[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. 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, Journal of Differential Equations, 185 (2002), 25–53. doi: 10.1006/jdeq.2001.4163. Google Scholar
[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. Google Scholar
[13]	J. K. Hale, Asmptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. Google Scholar
[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. Google Scholar
[15]	L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2005. doi: 10.1007/b138375. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23–100. doi: 10.4171/rmi/1049. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. suppl., 2005 (2005), 824-832. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. C. Evans, Partial Differential Equations, American Mathematical Soc., Providence, RI, 2010. doi: 10.1090/GSM/019. Google Scholar
[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. 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, Journal of Differential Equations, 185 (2002), 25–53. doi: 10.1006/jdeq.2001.4163. Google Scholar
[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. Google Scholar
[13]	J. K. Hale, Asmptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025. Google Scholar
[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. Google Scholar
[15]	L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer, 2005. doi: 10.1007/b138375. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), 23–100. doi: 10.4171/rmi/1049. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. suppl., 2005 (2005), 824-832. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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