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March 2019, 18(2): 911-930. doi: 10.3934/cpaa.2019044

## Attractors and their stability on Boussinesq type equations with gentle dissipation

 1 School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China 2 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

* Corresponding author

Received  June 2018 Revised  July 2018 Published  October 2018

Fund Project: The authors are supported by NNSF of China (No. 11671367)

The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:$u_{tt}+Δ^2 u+(-Δ)^{α} u_{t}-Δ f(u) = g(x)$, with $α∈ (0, 1)$. For general bounded domain $Ω\subset \mathbb{R}^N (N≥1)$, we show that there exists a critical exponent $p_α\equiv\frac{N+2(2α-1)}{(N-2)^+}$ depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: $1≤p <p_α$, (ⅰ) the weak solutions of the equations are of additionally global smoothness when $t>0$; (ⅱ) the related dynamical system possesses a global attractor $\mathcal{A}_α$ and an exponential attractor $\mathcal{A}^α_{exp}$ in natural energy space for each $α∈ (0, 1)$, respectively; (ⅲ) the family of global attractors $\{\mathcal{A}_α\}$ is upper semicontinuous at each point $α_0∈ (0,1]$, i.e., for any neighborhood U of $\mathcal{A}_{α_0}, \mathcal{A}_α\subset U$ when $|α-α_0|\ll 1$. These results extend those for structural damping case: $α∈ [1, 2)$ in [31,32].

Citation: Zhijian Yang, Pengyan Ding, Xiaobin Liu. Attractors and their stability on Boussinesq type equations with gentle dissipation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 911-930. doi: 10.3934/cpaa.2019044
##### References:
 [1] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Continuous Dynam. Systems - A, 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. [2] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29. [3] E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, J. Evol. Equ., (2018). [4] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982), 433-454. doi: 10.1090/qam/644099. [5] S. P. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems, Lecture Notes in Math., 1354 (1988), Springer-Verlag, 234-256. doi: 10.1007/BFb0089601. [6] S. P. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. [7] S. P. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1/2, Proceedings of AMS, 110 (1990), 401-415. doi: 10.2307/2048084. [8] Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Discrete Continuous Dynam. Systems - A, 17 (2007), 691-711. doi: 10.3934/dcds.2007.17.691. [9] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative System, Typography, layout, ACTA, 2002. [10] I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Continuous Dynam. Systems -A, 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777. [11] I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Communications in Partial Differential Equations, 36 (2010), 67-99. doi: 10.1080/03605302.2010.484472. [12] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, 195 (2008). doi: 10.1090/memo/0912. [13] I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. [14] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. [15] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4. [16] P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628. doi: 10.1002/cpa.3160350502. [17] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186. [18] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Continuous Dynam. Systems - A, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211. [19] S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. R. Soc. Edinb., 140A (2010), 329-366. doi: 10.1017/S0308210509000365. [20] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010. [21] L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117. [22] K. Li and S. H. Fu, Asymptotic behavior for the damped Boussinesq equation with critical nonlinearity, Appl. Math. Lett., 30 (2014), 44-50. doi: 10.1016/j.aml.2013.12.010. [23] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106 (1993), 257-293. doi: 10.1006/jdeq.1993.1108. [24] A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. [25] A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Analysis, 87 (2014), 191-221. [26] J. Simon, Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360. [27] V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations, 10 (1997), 1197-1211. [28] V. Varlamov, Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete Continuous Dynam. Systems - A, 7 (2001), 675-702. doi: 10.3934/dcds.2001.7.675. [29] V. Varlamov and A. Balogh, Forced nonlinear oscillations of elastic membranes, Nonlinear Anal. RWA., 7 (2006), 1005-1028. doi: 10.1016/j.nonrwa.2005.09.006. [30] S. B. Wang and X. Su, Global existence and long-time behavior of the initial-boundary value problem for the dissipative Boussinesq equation, Nonlinear Anal. RWA, 31 (2016), 552-568. doi: 10.1016/j.nonrwa.2016.03.002. [31] Z. J. Yang, Longtime dynamics of the damped Boussinesq equation, J. Math. Anal. Appl., 399 (2013), 180-190. doi: 10.1016/j.jmaa.2012.09.042. [32] Z. J. Yang and P. Y. Ding, Longtime dynamics of Boussinesq type equations with fractional damping, Nonlinear Analysis, 161 (2017), 108-130. doi: 10.1016/j.na.2017.05.015. [33] Z. J. Yang, P. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510. doi: 10.1016/j.jmaa.2016.04.079. [34] Z. J. Yang, Z. M. Liu and P. P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Commun. Contemp. Math., 18 (2016), 155055. doi: 10.1142/S0219199715500558. [35] Z. J. Yang, Z. M. Liu and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Continuous Dynam. Systems - A, 36 (2016), 6557-6580. doi: 10.3934/dcds.2016084. [36] Z. J. Yang and Z. M. Liu, Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities, Nonlinearity, 30 (2017), 1120-1145. doi: 10.1088/1361-6544/aa599f.

show all references

##### References:
 [1] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Continuous Dynam. Systems - A, 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. [2] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29. [3] E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, J. Evol. Equ., (2018). [4] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982), 433-454. doi: 10.1090/qam/644099. [5] S. P. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems, Lecture Notes in Math., 1354 (1988), Springer-Verlag, 234-256. doi: 10.1007/BFb0089601. [6] S. P. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. [7] S. P. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1/2, Proceedings of AMS, 110 (1990), 401-415. doi: 10.2307/2048084. [8] Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Discrete Continuous Dynam. Systems - A, 17 (2007), 691-711. doi: 10.3934/dcds.2007.17.691. [9] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative System, Typography, layout, ACTA, 2002. [10] I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Continuous Dynam. Systems -A, 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777. [11] I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Communications in Partial Differential Equations, 36 (2010), 67-99. doi: 10.1080/03605302.2010.484472. [12] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, 195 (2008). doi: 10.1090/memo/0912. [13] I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. [14] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. [15] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4. [16] P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628. doi: 10.1002/cpa.3160350502. [17] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186. [18] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Continuous Dynam. Systems - A, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211. [19] S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. R. Soc. Edinb., 140A (2010), 329-366. doi: 10.1017/S0308210509000365. [20] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010. [21] L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117. [22] K. Li and S. H. Fu, Asymptotic behavior for the damped Boussinesq equation with critical nonlinearity, Appl. Math. Lett., 30 (2014), 44-50. doi: 10.1016/j.aml.2013.12.010. [23] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106 (1993), 257-293. doi: 10.1006/jdeq.1993.1108. [24] A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. [25] A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Analysis, 87 (2014), 191-221. [26] J. Simon, Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360. [27] V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations, 10 (1997), 1197-1211. [28] V. Varlamov, Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete Continuous Dynam. Systems - A, 7 (2001), 675-702. doi: 10.3934/dcds.2001.7.675. [29] V. Varlamov and A. Balogh, Forced nonlinear oscillations of elastic membranes, Nonlinear Anal. RWA., 7 (2006), 1005-1028. doi: 10.1016/j.nonrwa.2005.09.006. [30] S. B. Wang and X. Su, Global existence and long-time behavior of the initial-boundary value problem for the dissipative Boussinesq equation, Nonlinear Anal. RWA, 31 (2016), 552-568. doi: 10.1016/j.nonrwa.2016.03.002. [31] Z. J. Yang, Longtime dynamics of the damped Boussinesq equation, J. Math. Anal. Appl., 399 (2013), 180-190. doi: 10.1016/j.jmaa.2012.09.042. [32] Z. J. Yang and P. Y. Ding, Longtime dynamics of Boussinesq type equations with fractional damping, Nonlinear Analysis, 161 (2017), 108-130. doi: 10.1016/j.na.2017.05.015. [33] Z. J. Yang, P. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510. doi: 10.1016/j.jmaa.2016.04.079. [34] Z. J. Yang, Z. M. Liu and P. P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Commun. Contemp. Math., 18 (2016), 155055. doi: 10.1142/S0219199715500558. [35] Z. J. Yang, Z. M. Liu and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Continuous Dynam. Systems - A, 36 (2016), 6557-6580. doi: 10.3934/dcds.2016084. [36] Z. J. Yang and Z. M. Liu, Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities, Nonlinearity, 30 (2017), 1120-1145. doi: 10.1088/1361-6544/aa599f.
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