\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Attractors and their stability on Boussinesq type equations with gentle dissipation

  • * Corresponding author

    * Corresponding author 
The authors are supported by NNSF of China (No. 11671367)
Abstract Full Text(HTML) Related Papers Cited by
  • 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].

    Mathematics Subject Classification: Primary: 35B40, 35B41, 35B65; Secondary: 37L30, 37L15.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   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.
      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. 
      E. Cerpa  and  I. Rivas , On the controllability of the Boussinesq equation in low regularity, J. Evol. Equ., (2018) . 
      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.
      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.
      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. 
      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.
      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.
      I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative System, Typography, layout, ACTA, 2002.
      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.
      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.
      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.
      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. 
      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.
      I. ChueshovDynamics of Quasi-Stable Dissipative Systems, Springer, 2015.  doi: 10.1007/978-3-319-22903-4.
      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.
      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.
      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.
      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.
      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.
      L. V. Kapitanski  and  I. N. Kostin , Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991) , 97-117. 
      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.
      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.
      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. 
      A. Savostianov  and  S. Zelik , Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Analysis, 87 (2014) , 191-221. 
      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.
      V. Varlamov , On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations, 10 (1997) , 1197-1211. 
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
  • 加载中
SHARE

Article Metrics

HTML views(321) PDF downloads(243) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return