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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.  Google Scholar

[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.   Google Scholar

[3]

E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, J. Evol. Equ., (2018).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative System, Typography, layout, ACTA, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.  doi: 10.1007/978-3-319-22903-4.  Google Scholar
[16]

P. DeiftC. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.  doi: 10.1002/cpa.3160350502.  Google Scholar

[17]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.  Google Scholar

[18]

P. FabrieC. GalusinskiA. 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.  Google Scholar

[19]

S. GattiA. MiranvilleV. 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.  Google Scholar

[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.  Google Scholar

[21]

L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Analysis, 87 (2014), 191-221.   Google Scholar

[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.  Google Scholar

[27]

V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations, 10 (1997), 1197-1211.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Z. J. YangP. 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.  Google Scholar

[34]

Z. J. YangZ. 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.  Google Scholar

[35]

Z. J. YangZ. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[3]

E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, J. Evol. Equ., (2018).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative System, Typography, layout, ACTA, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.  doi: 10.1007/978-3-319-22903-4.  Google Scholar
[16]

P. DeiftC. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.  doi: 10.1002/cpa.3160350502.  Google Scholar

[17]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.  Google Scholar

[18]

P. FabrieC. GalusinskiA. 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.  Google Scholar

[19]

S. GattiA. MiranvilleV. 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.  Google Scholar

[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.  Google Scholar

[21]

L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Analysis, 87 (2014), 191-221.   Google Scholar

[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.  Google Scholar

[27]

V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations, 10 (1997), 1197-1211.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Z. J. YangP. 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.  Google Scholar

[34]

Z. J. YangZ. 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.  Google Scholar

[35]

Z. J. YangZ. 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.  Google Scholar

[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.  Google Scholar

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