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On a Kirchhoff wave model with nonlocal nonlinear damping
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Moving and oblique observations of beams and plates
Robust attractors for a Kirchhoff-Boussinesq type equation
1. | School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China |
2. | College of Science, Zhongyuan University of Technology, No.41, Zhongyuan Road, Zhengzhou 450007, China |
The paper studies the existence of the pullback attractors and robust pullback exponential attractors for a Kirchhoff-Boussinesq type equation: $ u_{tt}-\Delta u_{t}+\Delta^{2} u = div\Big\{\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Big\}+\Delta g(u)+f(x,t) $. We show that when the growth exponent $ p $ of the nonlinearity $ g(u) $ is up to the critical range: $ 1\leq p\leq p^*\equiv\frac{N+2}{(N-2)^{+}} $, (ⅰ) the IBVP of the equation is well-posed, and its solution has additionally global regularity when $ t>\tau $; (ⅱ) the related dynamical process $ \{U_f(t,\tau)\} $ has a pullback attractor; (ⅲ) in particular, when $ 1\leq p< p^* $, the process $ \{U_f(t,\tau)\} $ has a family of pullback exponential attractors, which is stable with respect to the perturbation $ f\in \Sigma $ (the sign space).
References:
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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.
doi: 10.1007/BF01218475. |
[2] |
A. N. Carvalho, I. A. Langa and J. C. Robinson,
On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824.
doi: 10.1016/j.na.2009.01.016. |
[3] |
A. N. Carvalho, I. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[4] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: Theoretical result, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.
doi: 10.3934/cpaa.2013.12.3047. |
[5] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.
doi: 10.3934/cpaa.2014.13.1141. |
[6] |
I. Chueshov and I. Lasiecka,
Existence, uniqueness of weak solutions and global attractors for a class of 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.
doi: 10.3934/dcds.2006.15.777. |
[7] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008).
doi: 10.1090/memo/0912. |
[8] |
I. Chueshov and I. Lasiecka,
On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations, 36 (2011), 67-69.
doi: 10.1080/03605302.2010.484472. |
[9] |
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. |
[10] |
R. Czaja and M. A. Efendiev,
Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.
doi: 10.1016/j.jmaa.2011.03.053. |
[11] |
P. Y. Ding, Z. J. Yang and Y. N. Li,
Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Letters, 76 (2018), 40-45.
doi: 10.1016/j.aml.2017.07.008. |
[12] |
M. A. Efendiev, A. Miranville and S. Zelik,
Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[13] |
M. A. Efendiev, Y. Yamamoto and A. Yagi,
Exponential attractors for nonautonomous dynamical systems, J. Math. Soc. Japan, 63 (2011), 647-673.
doi: 10.2969/jmsj/06320647. |
[14] |
M. Grasselli, G. Schimperna and S. Zelik,
On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170.
doi: 10.1080/03605300802608247. |
[15] |
M. Grasselli, G. Schimperna, A. Segatti and S. Zelik,
On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404.
doi: 10.1007/s00028-009-0017-7. |
[16] |
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. |
[17] |
S. Kawashima and Y. Shibata,
Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys., 148 (1992), 189-208.
doi: 10.1007/BF02102372. |
[18] |
K. Kobayashi, H. Pecher and Y. Shibata,
On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Math. Ann., 296 (1993), 215-234.
doi: 10.1007/BF01445103. |
[19] |
L. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
doi: 10.3934/dcds.2010.26.1329. |
[20] |
J. Lagnese and J. L. Lions, Modeling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6. Masson, Paris, 1988. |
[21] |
J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
[22] |
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. |
[23] |
T. F. Ma and M. L. Pelicer,
Attractors for weakly damped beam equation with $p$-Laplacian, Discrete Contin. Dyn. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., 34 (2013), 525-534.
doi: 10.3934/proc.2013.2013.525. |
[24] |
M. Nakao,
Energy decay for the quasilinear wave equation with viscosity, Math. Z., 219 (1995), 289-299.
doi: 10.1007/BF02572366. |
[25] |
J. Simon,
Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[26] |
C. Y. Sun, D. M. Cao and J. Q. Duan,
Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.
doi: 10.1088/0951-7715/19/11/008. |
[27] |
V. Varlamov,
Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete Contin. Dynam. Systems, 7 (2001), 675-702.
doi: 10.3934/dcds.2001.7.675. |
[28] |
Y. H. Wang and C. K. Zhong,
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209.
doi: 10.3934/dcds.2013.33.3189. |
[29] |
Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 (2010), 092703, 25 pp.
doi: 10.1063/1.3477939. |
[30] |
Z. J. Yang,
Longtime dynamics of the damped Boussinesq equations, J. Math. Anal. Appl., 399 (2013), 180-190.
doi: 10.1016/j.jmaa.2012.09.042. |
[31] |
Z. J. Yang and Z. M. Liu,
Longtime dynamics of the for the quasi-linear wave equations with structural damping and supercritical nonlinearity, Nonlinearity, 30 (2017), 1120-1145.
doi: 10.1088/1361-6544/aa599f. |
[32] |
Z. J. Yang and P. Y. Ding,
Longtime dynamics of Boussinesq type equations with fractional damping, Nonlinear Anal., 161 (2017), 108-130.
doi: 10.1016/j.na.2017.05.015. |
[33] |
Z. J. Yang and Y. N. Li,
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653.
doi: 10.3934/dcds.2018111. |
[34] |
Z. J. Yang, P. Y. Ding and X. B. Liu,
Attractors and their stability on Boussinesq type equations with gentle dissipation, Comm. Pure Appl. Anal., 18 (2019), 911-930.
doi: 10.3934/cpaa.2019044. |
[35] |
X. G. Yang, Z. H. Fan and K. Li,
Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity, Math. Meth. Appl. Sci., 39 (2016), 3075-3087.
doi: 10.1002/mma.3753. |
show all references
References:
[1] |
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.
doi: 10.1007/BF01218475. |
[2] |
A. N. Carvalho, I. A. Langa and J. C. Robinson,
On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824.
doi: 10.1016/j.na.2009.01.016. |
[3] |
A. N. Carvalho, I. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[4] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: Theoretical result, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.
doi: 10.3934/cpaa.2013.12.3047. |
[5] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.
doi: 10.3934/cpaa.2014.13.1141. |
[6] |
I. Chueshov and I. Lasiecka,
Existence, uniqueness of weak solutions and global attractors for a class of 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.
doi: 10.3934/dcds.2006.15.777. |
[7] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008).
doi: 10.1090/memo/0912. |
[8] |
I. Chueshov and I. Lasiecka,
On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations, 36 (2011), 67-69.
doi: 10.1080/03605302.2010.484472. |
[9] |
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. |
[10] |
R. Czaja and M. A. Efendiev,
Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.
doi: 10.1016/j.jmaa.2011.03.053. |
[11] |
P. Y. Ding, Z. J. Yang and Y. N. Li,
Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Letters, 76 (2018), 40-45.
doi: 10.1016/j.aml.2017.07.008. |
[12] |
M. A. Efendiev, A. Miranville and S. Zelik,
Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[13] |
M. A. Efendiev, Y. Yamamoto and A. Yagi,
Exponential attractors for nonautonomous dynamical systems, J. Math. Soc. Japan, 63 (2011), 647-673.
doi: 10.2969/jmsj/06320647. |
[14] |
M. Grasselli, G. Schimperna and S. Zelik,
On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170.
doi: 10.1080/03605300802608247. |
[15] |
M. Grasselli, G. Schimperna, A. Segatti and S. Zelik,
On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404.
doi: 10.1007/s00028-009-0017-7. |
[16] |
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. |
[17] |
S. Kawashima and Y. Shibata,
Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys., 148 (1992), 189-208.
doi: 10.1007/BF02102372. |
[18] |
K. Kobayashi, H. Pecher and Y. Shibata,
On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Math. Ann., 296 (1993), 215-234.
doi: 10.1007/BF01445103. |
[19] |
L. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
doi: 10.3934/dcds.2010.26.1329. |
[20] |
J. Lagnese and J. L. Lions, Modeling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6. Masson, Paris, 1988. |
[21] |
J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
[22] |
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. |
[23] |
T. F. Ma and M. L. Pelicer,
Attractors for weakly damped beam equation with $p$-Laplacian, Discrete Contin. Dyn. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., 34 (2013), 525-534.
doi: 10.3934/proc.2013.2013.525. |
[24] |
M. Nakao,
Energy decay for the quasilinear wave equation with viscosity, Math. Z., 219 (1995), 289-299.
doi: 10.1007/BF02572366. |
[25] |
J. Simon,
Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[26] |
C. Y. Sun, D. M. Cao and J. Q. Duan,
Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.
doi: 10.1088/0951-7715/19/11/008. |
[27] |
V. Varlamov,
Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete Contin. Dynam. Systems, 7 (2001), 675-702.
doi: 10.3934/dcds.2001.7.675. |
[28] |
Y. H. Wang and C. K. Zhong,
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209.
doi: 10.3934/dcds.2013.33.3189. |
[29] |
Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 (2010), 092703, 25 pp.
doi: 10.1063/1.3477939. |
[30] |
Z. J. Yang,
Longtime dynamics of the damped Boussinesq equations, J. Math. Anal. Appl., 399 (2013), 180-190.
doi: 10.1016/j.jmaa.2012.09.042. |
[31] |
Z. J. Yang and Z. M. Liu,
Longtime dynamics of the for the quasi-linear wave equations with structural damping and supercritical nonlinearity, Nonlinearity, 30 (2017), 1120-1145.
doi: 10.1088/1361-6544/aa599f. |
[32] |
Z. J. Yang and P. Y. Ding,
Longtime dynamics of Boussinesq type equations with fractional damping, Nonlinear Anal., 161 (2017), 108-130.
doi: 10.1016/j.na.2017.05.015. |
[33] |
Z. J. Yang and Y. N. Li,
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653.
doi: 10.3934/dcds.2018111. |
[34] |
Z. J. Yang, P. Y. Ding and X. B. Liu,
Attractors and their stability on Boussinesq type equations with gentle dissipation, Comm. Pure Appl. Anal., 18 (2019), 911-930.
doi: 10.3934/cpaa.2019044. |
[35] |
X. G. Yang, Z. H. Fan and K. Li,
Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity, Math. Meth. Appl. Sci., 39 (2016), 3075-3087.
doi: 10.1002/mma.3753. |
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