• Previous Article
    Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation
  • DCDS Home
  • This Issue
  • Next Article
    Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation
July  2013, 33(7): 3189-3209. doi: 10.3934/dcds.2013.33.3189

Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models

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.
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.

[1]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[2]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

[3]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653

[4]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252

[5]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[6]

Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure and Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023

[7]

Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116

[8]

Wenlong Sun. The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28 (3) : 1343-1356. doi: 10.3934/era.2020071

[9]

Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 73-108. doi: 10.3934/dcds.2021108

[10]

Tingting Liu, Qiaozhen Ma, Ling Xu. Attractor of the Kirchhoff type plate equation with memory and nonlinear damping on the whole time-dependent space. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022046

[11]

Shengfan Zhou, Caidi Zhao, Yejuan Wang. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1259-1277. doi: 10.3934/dcds.2008.21.1259

[12]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115

[13]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035

[14]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[15]

Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139

[16]

Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010

[17]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[18]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[19]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[20]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (118)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]