June  2020, 9(2): 469-486. doi: 10.3934/eect.2020020

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

* Corresponding author: Zhijian Yang

Received  October 2018 Revised  January 2019 Published  June 2020 Early access  December 2019

Fund Project: The authors are supported by NNSF of China (Grant No. 11671367)

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).

Citation: Zhijian Yang, Na Feng, Yanan Li. Robust attractors for a Kirchhoff-Boussinesq type equation. Evolution Equations and Control Theory, 2020, 9 (2) : 469-486. doi: 10.3934/eect.2020020
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. CarvalhoI. 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.

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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.

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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.

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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. DingZ. 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. EfendievA. 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. EfendievY. 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. GrasselliG. 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. GrasselliG. SchimpernaA. 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. KobayashiH. 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. LangaA. 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. SunD. 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. YangP. 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. YangZ. 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. CarvalhoI. 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. DingZ. 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. EfendievA. 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. EfendievY. 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. GrasselliG. 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. GrasselliG. SchimpernaA. 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. KobayashiH. 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. LangaA. 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. SunD. 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. YangP. 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. YangZ. 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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