• Previous Article
    Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation
  • CPAA Home
  • This Issue
  • Next Article
    Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers
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 and 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. DeiftC. 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. 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.

[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.

[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.

[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. 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.

[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.

[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.

[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. DeiftC. 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. 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.

[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.

[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.

[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. 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.

[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.

[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.

[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.

[1]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

[2]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1

[3]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[4]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078

[5]

Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations and Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57

[6]

Huy Tuan Nguyen, Nguyen Anh Tuan, Chao Yang. Global well-posedness for fractional Sobolev-Galpern type equations. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2637-2665. doi: 10.3934/dcds.2021206

[7]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101

[8]

Xiaoxiao Suo, Quansen Jiu. Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022063

[9]

Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113

[10]

Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114

[11]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[12]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653

[13]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic and Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395

[14]

Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure and Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185

[15]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292

[16]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[17]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[18]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[19]

Adalet Hanachi, Haroune Houamed, Mohamed Zerguine. On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6473-6506. doi: 10.3934/dcds.2020287

[20]

Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1613-1650. doi: 10.3934/dcds.2019072

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (230)
  • HTML views (195)
  • Cited by (3)

Other articles
by authors

[Back to Top]