Robust attractors for a Kirchhoff-Boussinesq type equation

Zhijian Yang; Na Feng; Yanan Li

doi:10.3934/eect.2020020

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

Robust attractors for a Kirchhoff-Boussinesq type equation

Zhijian Yang ^1,, , Na Feng ^2, and Yanan Li ^1,

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).

Keywords: Kirchhoff-Boussinesq type equation, well-posedness, pullback attractor, pullback exponential attractor, stability.

Mathematics Subject Classification: Primary: 35B40, 37L15, 37B55; Secondary: 35B41, 35B33, 35B65.

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

References:

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[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. Google Scholar
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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	M. Nakao, Energy decay for the quasilinear wave equation with viscosity, Math. Z., 219 (1995), 289-299. doi: 10.1007/BF02572366. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 (2010), 092703, 25 pp. doi: 10.1063/1.3477939. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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