# American Institute of Mathematical Sciences

December  2021, 26(12): 6267-6284. doi: 10.3934/dcdsb.2021018

## Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models

 1 College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China 2 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China 3 College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

* Corresponding author: Zhijian Yang

Received  September 2020 Revised  November 2020 Published  December 2021 Early access  January 2021

Fund Project: The authors are supported by NSFC (Grant No. 11671367)

The paper investigates the existence and the continuity of uniform attractors for the non-autonomous Kirchhoff wave equations with strong damping: $u_{tt}-(1+\epsilon\|\nabla u\|^{2})\Delta u-\Delta u_{t}+f(u) = g(x,t)$, where $\epsilon\in [0,1]$ is an extensibility parameter. It shows that when the nonlinearity $f(u)$ is of optimal supercritical growth $p: \frac{N+2}{N-2} = p^* Citation: Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6267-6284. doi: 10.3934/dcdsb.2021018 ##### References:  [1] A. V. Babin and S. Yu. Pilyugin, Continuous dependence of attractors on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310. doi: 10.1007/BF02355582. [2] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. [4] 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. [5] P. Y. Ding, Z. 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. [6] 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. [7] J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326. doi: 10.1007/BF01790353. [8] L. T. Hoang, E. J. Olason and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Sc., 143 (2015), 4389-4395. doi: 10.1090/proc/12598. [9] L. T. Hoang, E. J. Olason and J. C. Robinson, Continuity of pullback and uniform attractors, J. Differential Equations, 264 (2018), 4067-4093. doi: 10.1016/j.jde.2017.12.002. [10] G. Kirchhoff, Vorlesungen über Mechanik, Lectures on Mechanics, Teubner, Stuttgart, 1883. [11] Y. N. Li and Z. J. Yang, Robustness of attractors for non-autonomous Kirchhoff wave models with strong nonlinear damping, Appl. Math. Optim., (2019). doi: 10.1007/s00245-019-09644-4. [12] S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for non-autonomous$2D$Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. [13] H. L. Ma and C. K. Zhong, Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133. doi: 10.1016/j.aml.2017.06.002. [14] H. L. Ma, J. Zhang and C. K. Zhong, Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping, Discrete Contin. Dyn. Syst.-B, (2019). doi: 10.3934/dcdsb.2019027. [15] H. L. Ma, J. Zhang and C. K. Zhong, Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension, J. Math. Anal. Appl., 484 (2020), 123670, 15 pp. doi: 10.1016/j.jmaa.2019.123670. [16] 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. [17] I. Moise, R. Rosa and X. Wang, Attractors for noncompact non-autonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473. [18] 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. [19] 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. [20] J. Simon, Compact sets in the space$L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360. [21] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996. [22] C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318. doi: 10.1137/060663805. [23] B. X. Wang, Uniform attractors of non-autonomous discrete reaction-diffusion systems in weighted spaces, Int. J. Bifurcation Chaos, 18 (2008), 659-716. doi: 10.1142/S0218127408020598. [24] Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189. [25] 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. [26] Z. J. Yang and P. Y. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on$\mathbb{R}^N$, J. Math. Anal. Appl., 434 (2016), 1826-1851. doi: 10.1016/j.jmaa.2015.10.013. [27] X.-G. Yang, Marcelo J. D. Nascimento and L. Pelicer Maurício, Uniform attractors for non-autonomous plate equations with$p$-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961. doi: 10.3934/dcds.2020100. [28] S. Zelik, Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces, Discrete Contin. Dyn. Syst.-B, 20 (2015), 781-810. doi: 10.3934/dcdsb.2015.20.781. show all references ##### References:  [1] A. V. Babin and S. Yu. Pilyugin, Continuous dependence of attractors on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310. doi: 10.1007/BF02355582. [2] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. [4] 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. [5] P. Y. Ding, Z. 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. [6] 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. [7] J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326. doi: 10.1007/BF01790353. [8] L. T. Hoang, E. J. Olason and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Sc., 143 (2015), 4389-4395. doi: 10.1090/proc/12598. [9] L. T. Hoang, E. J. Olason and J. C. Robinson, Continuity of pullback and uniform attractors, J. Differential Equations, 264 (2018), 4067-4093. doi: 10.1016/j.jde.2017.12.002. [10] G. Kirchhoff, Vorlesungen über Mechanik, Lectures on Mechanics, Teubner, Stuttgart, 1883. [11] Y. N. Li and Z. J. Yang, Robustness of attractors for non-autonomous Kirchhoff wave models with strong nonlinear damping, Appl. Math. Optim., (2019). doi: 10.1007/s00245-019-09644-4. [12] S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for non-autonomous$2D$Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. [13] H. L. Ma and C. K. Zhong, Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133. doi: 10.1016/j.aml.2017.06.002. [14] H. L. Ma, J. Zhang and C. K. Zhong, Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping, Discrete Contin. Dyn. Syst.-B, (2019). doi: 10.3934/dcdsb.2019027. [15] H. L. Ma, J. Zhang and C. K. Zhong, Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension, J. Math. Anal. Appl., 484 (2020), 123670, 15 pp. doi: 10.1016/j.jmaa.2019.123670. [16] 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. [17] I. Moise, R. Rosa and X. Wang, Attractors for noncompact non-autonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473. [18] 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. [19] 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. [20] J. Simon, Compact sets in the space$L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360. [21] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996. [22] C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318. doi: 10.1137/060663805. [23] B. X. Wang, Uniform attractors of non-autonomous discrete reaction-diffusion systems in weighted spaces, Int. J. Bifurcation Chaos, 18 (2008), 659-716. doi: 10.1142/S0218127408020598. [24] Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189. [25] 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. [26] Z. J. Yang and P. Y. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on$\mathbb{R}^N$, J. Math. Anal. Appl., 434 (2016), 1826-1851. doi: 10.1016/j.jmaa.2015.10.013. [27] X.-G. Yang, Marcelo J. D. Nascimento and L. Pelicer Maurício, Uniform attractors for non-autonomous plate equations with$p$-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961. doi: 10.3934/dcds.2020100. [28] S. Zelik, Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces, Discrete Contin. Dyn. Syst.-B, 20 (2015), 781-810. doi: 10.3934/dcdsb.2015.20.781.  [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] Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743 [3] Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. 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