American Institute of Mathematical Sciences

• Previous Article
Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation
• DCDS Home
• This Issue
• Next Article
Persistence properties and infinite propagation for the modified 2-component Camassa--Holm 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 & Continuous Dynamical Systems - A, 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. doi: 10.1080/03605309208820866. Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [16] M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type,, Math. Meth. Appl. Sci., 22 (1999), 375. 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, (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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2005.05.031. Google Scholar [24] G. Kirchhoff, "Vorlesungen über Mechanik,", Teubner, (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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [34] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. 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. 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. 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. 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). doi: 10.1063/1.3277152. Google Scholar [39] Y. H. Wang, Pullback attractors for nonautonomous wave equations with critical exponent,, Nonlinear Anal., 68 (2008), 365. 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. 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. 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. doi: 10.1080/03605309208820866. Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [16] M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type,, Math. Meth. Appl. Sci., 22 (1999), 375. 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, (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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2005.05.031. Google Scholar [24] G. Kirchhoff, "Vorlesungen über Mechanik,", Teubner, (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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [34] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. 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. 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. 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. 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). doi: 10.1063/1.3277152. Google Scholar [39] Y. H. Wang, Pullback attractors for nonautonomous wave equations with critical exponent,, Nonlinear Anal., 68 (2008), 365. 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. 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. doi: 10.1016/j.jde.2007.08.004. Google Scholar
 [1] Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & 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 & 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 & Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653 [4] Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899 [5] Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023 [6] Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116 [7] Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139 [8] Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010 [9] Shengfan Zhou, Caidi Zhao, Yejuan Wang. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1259-1277. doi: 10.3934/dcds.2008.21.1259 [10] Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035 [11] Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120 [12] Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115 [13] Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195 [14] T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037 [15] Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060 [16] Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95 [17] Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91 [18] Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111 [19] Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721 [20] Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009

2018 Impact Factor: 1.143