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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

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