Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models

Yonghai Wang; Chengkui Zhong

doi:10.3934/dcds.2013.33.3189

Article Contents

2013, Volume 33, Issue 7: 3189-3209. Doi: 10.3934/dcds.2013.33.3189

This issue Previous Article Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation Next Article Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation

Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models

Yonghai Wang^1, and
Chengkui Zhong^2,

1.
Department of Applied Mathematics, Donghua University, Shanghai, Songjiang, 201620
2.
Department of Mathematics, Nanjing University, Nanjing 210093

Received: March 2012

Revised: August 2012

Published: July 2013

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Abstract

In this paper, we consider the upper semicontinuity of pullback attractors for a nonautonomous Kirchhoff wave model with strong damping. For this purpose, some necessary abstract results are established.

Keywords:

Mathematics Subject Classification: Primary: 37L05, 35B40; Secondary: 35B41.

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References

[1]	J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equations with critical exponents, Comm. Partial Differential Equations, 17 (1992), 841-866.doi: 10.1080/03605309208820866.
[2]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.
[3]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.doi: 10.1016/j.na.2009.09.037.
[4]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.doi: 10.1016/j.na.2005.03.111.
[5]	T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst., 9 (2008), 525-539.doi: 10.3934/dcdsb.2008.9.525.
[6]	T. Caraballo, M. J. Garrido-Atienza and B. Schmalfß, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.doi: 10.3934/dcds.2008.21.415.
[7]	T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.
[8]	T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Commun. Partial Diff. Eqns., 23 (1998), 1557-1581.doi: 10.1080/03605309808821394.
[9]	A. N. Carvalho, J. 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.
[10]	A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 223 (2007), 622-653.doi: 10.1016/j.jde.2006.08.009.
[11]	I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equatins, 252 (2012), 1229-1262.doi: 10.1016/j.jde.2011.08.022.
[12]	I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86.doi: 10.1016/j.jde.2006.09.019.
[13]	I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping," Mem. Amer. Math. Soc., 195, 2008.
[14]	C. M. Elliot and I. N. Kostin, Lower semicontinuity of a nonhyperbolic attractor fro the viscous Cahn-Hilliard equation, Nonlinearity, 9 (1996), 678-702.doi: 10.1088/0951-7715/9/3/005.
[15]	F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics and Stochastics Reports, 59 (1996), 21-45.
[16]	M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7.
[17]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988.
[18]	J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.doi: 10.1016/0022-0396(88)90104-0.
[19]	J. K. Hale and G. Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dynam. Diff. Eq., 2 (1990), 19-67.doi: 10.1007/BF01047769.
[20]	L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117.
[21]	V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equation, 247 (2009), 1120-1155.doi: 10.1016/j.jde.2009.04.010.
[22]	A. Kh. Khanmamedov, Global attractors for wave equation with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.doi: 10.1016/j.jde.2006.06.001.
[23]	A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.doi: 10.1016/j.jmaa.2005.05.031.
[24]	G. Kirchhoff, "Vorlesungen über Mechanik," Teubner, Stuttgart, 1883.
[25]	P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.doi: 10.1016/j.jde.2007.10.031.
[26]	P. E. Kloeden, J. Real and C. Y. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.doi: 10.1016/j.jde.2008.11.017.
[27]	P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Roy. Soc. London A, 463 (2007), 163-181.doi: 10.1098/rspa.2006.1753.
[28]	S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371.doi: 10.1016/j.na.2009.01.187.
[29]	I. N. Kostin, Lower semicontinuity of a non-hyperbolic attractor, J. London Math. Soc., 52 (1995), 568-582.doi: 10.1112/jlms/52.3.568.
[30]	P. Marin-Rubio, A. M. Marquez-Duran and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Cont. Dyn. Syst., 31 (2011), 779-796.doi: 10.3934/dcds.2011.31.779.
[31]	T. Matsuyama and R. lkehata, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term, J. Math. Anal. Appl., 204 (1996), 729-753.doi: 10.1006/jmaa.1996.0464.
[32]	M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009), 652-659.doi: 10.1016/j.jmaa.2008.09.010.
[33]	M. Nakao and Z. J. Yang, Global attractors for some qusi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.
[34]	V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.doi: 10.1088/0951-7715/19/7/001.
[35]	J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.doi: 10.1016/S0022-0396(02)00038-4.
[36]	Y. J. Wang, C. K. Zhong and S. F. Zhou, Pullback attractors of nonautonomous dynamical systems, Discrete Contin. Dyn. Syst., 16 (2006), 587-614.doi: 10.3934/dcds.2006.16.705.
[37]	Y. H. Wang, On the upper semicontinuity of pullback attractors with applications to plate equations, Commun. Pure Appl. Anal., 9 (2010), 1653-1673.doi: 10.3934/cpaa.2010.9.1653.
[38]	Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations, J. Math. Phys., 51 (2010), 022701.doi: 10.1063/1.3277152.
[39]	Y. H. Wang, Pullback attractors for nonautonomous wave equations with critical exponent, Nonlinear Anal., TMA, 68 (2008), 365-376.doi: 10.1016/j.na.2006.11.002.
[40]	Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.
[41]	Z. J. Yang, Longtime behavior of the Kirchhoff type equation with strong damping in $\mathbbR^N$, J. Differential Equations, 242 (2007), 269-286.doi: 10.1016/j.jde.2007.08.004.