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  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^*<p<p^{**} = \frac{N+4}{(N-4)^+} $: (ⅰ) the related evolution process has in natural energy space $ \mathcal{H} = (H^1_0\cap L^{p+1})\times L^2 $ a compact uniform attractor $ \mathcal{A}^{\epsilon}_{\Sigma} $ for each $ \epsilon\in [0,1] $; (ⅱ) the family of compact uniform attractor $ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $ is continuous on $ \epsilon $ in a residual set $ I^*\subset [0,1] $ in the sense of $ \mathcal{H}_{ps} ( = (H^1_0\cap L^{p+1,w})\times L^2) $-topology; (ⅲ) $ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $ is upper semicontinuous on $ \epsilon\in [0,1] $ in $ \mathcal{H}_{ps} $-topology.

Citation: Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021018
References:
[1]

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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.  Google Scholar

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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

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[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.  Google Scholar

[21] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.   Google Scholar
[22]

C. Y. SunD. 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.  Google Scholar

[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.  Google Scholar

[24]

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[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.   Google Scholar

[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.  Google Scholar

[27]

X.-G. YangMarcelo 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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. T. HoangE. 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.  Google Scholar

[9]

L. T. HoangE. 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.  Google Scholar

[10]

G. Kirchhoff, Vorlesungen über Mechanik, Lectures on Mechanics, Teubner, Stuttgart, 1883. Google Scholar

[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.  Google Scholar

[12]

S. S. LuH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

I. MoiseR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.   Google Scholar
[22]

C. Y. SunD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

X.-G. YangMarcelo 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.  Google Scholar

[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.  Google Scholar

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