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Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers
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 |
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 [
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Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29.
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On the controllability of the Boussinesq equation in low regularity, J. Evol. Equ., (2018).
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G. Chen and D. L. Russell,
A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982), 433-454.
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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. |
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I. Chueshov,
Introduction to the Theory of Infinite-Dimensional Dissipative System, Typography, layout, ACTA, 2002. |
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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).
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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.
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[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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