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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 & 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.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping," Mem. Amer. Math. Soc., 195, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Kirchhoff, "Vorlesungen über Mechanik," Teubner, Stuttgart, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping," Mem. Amer. Math. Soc., 195, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Kirchhoff, "Vorlesungen über Mechanik," Teubner, Stuttgart, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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