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Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems
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 |
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.
References:
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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. |
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G. Kirchhoff, Vorlesungen über Mechanik, Lectures on Mechanics, Teubner, Stuttgart, 1883. Google Scholar |
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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. 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. |
[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. 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. |
[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. 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. |
[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. |
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