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July  2013, 33(7): 3189-3209. doi: 10.3934/dcds.2013.33.3189

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  January 2013

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: Pullback attractor, upper semicontinuity, Kirchhoff type..
Mathematics Subject Classification: Primary: 37L05, 35B40; Secondary: 35B4.
Citation: Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189
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 $\mathbb{R}^N2$, J. Differential Equations, 242 (2007), 269-286. doi: 10.1016/j.jde.2007.08.004.

show all references

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 $\mathbb{R}^N2$, J. Differential Equations, 242 (2007), 269-286. doi: 10.1016/j.jde.2007.08.004.

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