American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021018

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

Yanan Li 1, , Zhijian Yang 2,, and  Na Feng 3,

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.

Keywords: Non-autonomous Kirchhoff wave models, extensibility parameter, supercritical nonlinearity, uniform attractor, residual continuity, upper semicontinuity.
Mathematics Subject Classification: Primary: 37L15, 37L30; Secondary: 35B65, 35B33, 35B41.
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]

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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. 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.  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

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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. 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.  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

[1]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[2]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006
[3]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211
[4]

K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038
[5]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[6]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233
[7]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2725-3737. doi: 10.3934/dcds.2020383
[8]

Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266
[9]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252
[10]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[11]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043
[12]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[13]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060
[14]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019
[15]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057
[16]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400
[17]

Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254
[18]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
[19]

Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037
[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences