doi: 10.3934/dcdsb.2019036

Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations

School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou, 450001, China

* Corresponding author: Zhijian Yang

Received  March 2018 Revised  October 2018 Published  February 2019

Fund Project: This work is supported by National Natural Science Foundation of China (No.11671367)

The paper investigates the upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with structural damping: $ u_{tt}-M(\|\nabla u\|^2)\Delta u+(-\Delta)^\alpha u_t+f(u) = g(x,t) $, where $ \alpha\in(1/2, 1) $ is said to be a dissipative index. It shows that when the nonlinearity $ f(u) $ is of supercritical growth $ p: 1 \leq p< p_{\alpha}\equiv\frac{N+4\alpha}{(N-4\alpha)^+} $, the related evolution process has a pullback attractor for each $ \alpha\in(1/2, 1) $, and the family of pullback attractors is upper semicontinuous with respect to $ \alpha $. These results extend those in [27] for autonomous Kirchhoff wave models.

Citation: Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019036
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992.

[2]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Sys., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.

[3]

F. D. M. BezerraA. N. CarvalhoJ. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of dynamics, J. Math. Anal. Appl., 450 (2017), 377-405. doi: 10.1016/j.jmaa.2017.01.024.

[4]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283. doi: 10.1016/j.na.2010.11.032.

[5]

T. CaraballoG. 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.

[6]

A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Surez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.

[7]

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, 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009.

[8]

A. N. CarvalhoJ. 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.

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical System, Springer Science+Business, Media, LLC, 2013. doi: 10.1007/978-1-4614-4581-4.

[10]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.

[11]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022.

[12]

P. Y. DingZ. J. Yang and Y. N. Li, Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45. doi: 10.1016/j.aml.2017.07.008.

[13]

X. Fan and S. Zhou, Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type, Appl. Math. Comput., 158 (2004), 253-266. doi: 10.1016/j.amc.2003.08.147.

[14]

M. M. FreitasP. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945. doi: 10.1016/j.jde.2017.10.007.

[15]

K. Gabert and B. X. Wang, Non-autonomous attractors for singularly perturbed parabolic equation on $\mathbb{R}^n$, Nonlinear Anal., 73 (2010), 3336-3347. doi: 10.1016/j.na.2010.07.014.

[16]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92. doi: 10.1016/j.na.2013.06.008.

[17]

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.

[18]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010.

[19]

G. Kirchhoff, Vorlesungen über Mechanik, (German) [Lectures on Mechanics], Teubner, Stuttgart, 1883.

[20]

P. E. Kloeden, Pullback Attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632.

[21]

M. Nakao and Z. J. Yang, Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.

[22]

K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301. doi: 10.1006/jdeq.1997.3263.

[23]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177. doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0.

[24]

C. Y. SunD. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665. doi: 10.1088/0951-7715/19/11/008.

[25]

Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189.

[26]

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. doi: 10.1016/j.jde.2010.09.024.

[27]

Z. J. YangP. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510. doi: 10.1016/j.jmaa.2016.04.079.

[28]

Z. J. Yang and P. Y. Ding, Longtime dynamics of Boussinesq type equations with fractional damping, Nonlinear Anal., 161 (2017), 108-130. doi: 10.1016/j.na.2017.05.015.

[29]

Z. J. Yang and Y. N. Li, Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653. doi: 10.3934/dcds.2018111.

[30]

Z. J. Yang and Z. M. Liu, Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation, Appl. Math. Lett., 69 (2017), 22-28. doi: 10.1016/j.aml.2017.01.006.

[31]

Z. J. Yang and Z. M. Liu, Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation, J. Differential Equations, 264 (2018), 3976-4005. doi: 10.1016/j.jde.2017.11.035.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992.

[2]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Sys., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.

[3]

F. D. M. BezerraA. N. CarvalhoJ. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of dynamics, J. Math. Anal. Appl., 450 (2017), 377-405. doi: 10.1016/j.jmaa.2017.01.024.

[4]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283. doi: 10.1016/j.na.2010.11.032.

[5]

T. CaraballoG. 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.

[6]

A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Surez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.

[7]

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, 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009.

[8]

A. N. CarvalhoJ. 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.

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical System, Springer Science+Business, Media, LLC, 2013. doi: 10.1007/978-1-4614-4581-4.

[10]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.

[11]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022.

[12]

P. Y. DingZ. J. Yang and Y. N. Li, Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45. doi: 10.1016/j.aml.2017.07.008.

[13]

X. Fan and S. Zhou, Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type, Appl. Math. Comput., 158 (2004), 253-266. doi: 10.1016/j.amc.2003.08.147.

[14]

M. M. FreitasP. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945. doi: 10.1016/j.jde.2017.10.007.

[15]

K. Gabert and B. X. Wang, Non-autonomous attractors for singularly perturbed parabolic equation on $\mathbb{R}^n$, Nonlinear Anal., 73 (2010), 3336-3347. doi: 10.1016/j.na.2010.07.014.

[16]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92. doi: 10.1016/j.na.2013.06.008.

[17]

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.

[18]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010.

[19]

G. Kirchhoff, Vorlesungen über Mechanik, (German) [Lectures on Mechanics], Teubner, Stuttgart, 1883.

[20]

P. E. Kloeden, Pullback Attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632.

[21]

M. Nakao and Z. J. Yang, Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.

[22]

K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301. doi: 10.1006/jdeq.1997.3263.

[23]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177. doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0.

[24]

C. Y. SunD. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665. doi: 10.1088/0951-7715/19/11/008.

[25]

Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189.

[26]

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. doi: 10.1016/j.jde.2010.09.024.

[27]

Z. J. YangP. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510. doi: 10.1016/j.jmaa.2016.04.079.

[28]

Z. J. Yang and P. Y. Ding, Longtime dynamics of Boussinesq type equations with fractional damping, Nonlinear Anal., 161 (2017), 108-130. doi: 10.1016/j.na.2017.05.015.

[29]

Z. J. Yang and Y. N. Li, Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653. doi: 10.3934/dcds.2018111.

[30]

Z. J. Yang and Z. M. Liu, Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation, Appl. Math. Lett., 69 (2017), 22-28. doi: 10.1016/j.aml.2017.01.006.

[31]

Z. J. Yang and Z. M. Liu, Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation, J. Differential Equations, 264 (2018), 3976-4005. doi: 10.1016/j.jde.2017.11.035.

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