December  2017, 22(10): 3875-3890. doi: 10.3934/dcdsb.2017198

Robustness of exponentially κ-dissipative dynamical systems with perturbations

1. 

Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, China

2. 

Department of Applied Mathematics, Donghua University, Shanghai, 201620, China

3. 

Department of Mathematics, Nanjing University, Nanjing, 210093, China

* Corresponding author: Yonghai Wang

Received  September 2016 Revised  April 2017 Published  July 2017

We study the robustness of exponentially $κ$-dissipative dynamical systems with perturbed parameters $\varepsilon∈ E(\subset\mathbb{R})$. In particular, under some proper assumptions, we will construct a family of compact sets $\{\mathcal A_\varepsilon\}_{\varepsilon∈ E}$, which is positive invariant, uniformly exponentially attracting and equi-continuous. At last, an application to a Kirchhoff wave model is given.

Citation: Jin Zhang, Yonghai Wang, Chengkui Zhong. Robustness of exponentially κ-dissipative dynamical systems with perturbations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3875-3890. doi: 10.3934/dcdsb.2017198
References:
[1]

R. AraújoT. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087. doi: 10.1016/j.jde.2013.02.010. Google Scholar

[2]

A. CaixetaI. Lasiecka and V. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147. doi: 10.1016/j.jde.2016.03.006. Google Scholar

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A. CarvalhoJ. Cholewa and T. Dlotko, Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation, Proc. Roy. Soc. Edinburgh, 144 (2014), 13-51. doi: 10.1017/S0308210511001235. Google Scholar

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A. Carvalho and J. Cholewa, Exponential global attractors for semigroups in metric spaces with applications to differential equations, Ergod. Th. & Dynam. Sys., 31 (2011), 1641-1667. doi: 10.1017/S0143385710000702. Google Scholar

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A. CarvalhoJ. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer-Verlag, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

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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

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I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. Google Scholar

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K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar

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A. EdenC. FoiasB. Nicolaenko and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 559-562. Google Scholar

[10]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for Dissipative Evolution Equations, John Wiley & Sons, Chichester, 1994. Google Scholar

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M. EfendievA. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X. Google Scholar

[12]

M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation, Discrete Cont. Dyn. Syst., 35 (2015), 2539-2564. doi: 10.3934/dcds.2015.35.2539. Google Scholar

[13]

A. Khanmamedov, Strongly damped wave equation with exponential nonlinearities, J. Math. Anal. Appl., 419 (2014), 663-687. doi: 10.1016/j.jmaa.2014.05.010. Google Scholar

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P. KloedenJ. Real and C. Sun, Robust exponential attractors for non-autonomous equations with memory, Commun. Pure Appl. Anal., 10 (2011), 885-915. doi: 10.3934/cpaa.2011.10.885. Google Scholar

[15]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasgow Math. J., 46 (2004), 131-141. doi: 10.1017/S0017089503001605. Google Scholar

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A. MiranvilleV. Pata and S. Zelik, Exponential attractors for singular perturbed damped wave equations: A simple construction, Asymptotic Anal., 53 (2007), 1-12. Google Scholar

[17]

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

[18]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. Google Scholar

[19]

F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I, Russian Journal of Mathematical Physics, 15 (2008), 301-315. doi: 10.1134/S1061920808030014. Google Scholar

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A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Anal., 87 (2014), 191-221. doi: 10.3233/ASY-131208. Google Scholar

[21]

M. Silva and T. Ma, Longtime dynamics for a class of Kirchhoff models with memory, J. Math. Phy. , 54 (2013), 021505, 15pp. doi: 10.1063/1.4792606. Google Scholar

[22] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[23]

Y. Wang and C. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Cont. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189. Google Scholar

[24]

Z. Yang, Longtime behavior of the Kirchhoff type equation with strong damping on $\mathbb{R}^N$, J. Differential Equations, 242 (2007), 269-286. doi: 10.1016/j.jde.2007.08.004. Google Scholar

[25]

Z. Yang and Y. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278. doi: 10.1016/j.jde.2010.09.024. Google Scholar

[26]

J. ZhangP. KloedenM. Yang and C. Zhong, Global exponential $κ$-dissipative semigroups and exponential attraction, Discrete Cont. Dyn. Syst., 37 (2017), 3487-3502. doi: 10.3934/dcds.2017148. Google Scholar

show all references

References:
[1]

R. AraújoT. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087. doi: 10.1016/j.jde.2013.02.010. Google Scholar

[2]

A. CaixetaI. Lasiecka and V. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147. doi: 10.1016/j.jde.2016.03.006. Google Scholar

[3]

A. CarvalhoJ. Cholewa and T. Dlotko, Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation, Proc. Roy. Soc. Edinburgh, 144 (2014), 13-51. doi: 10.1017/S0308210511001235. Google Scholar

[4]

A. Carvalho and J. Cholewa, Exponential global attractors for semigroups in metric spaces with applications to differential equations, Ergod. Th. & Dynam. Sys., 31 (2011), 1641-1667. doi: 10.1017/S0143385710000702. Google Scholar

[5]

A. CarvalhoJ. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer-Verlag, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[6]

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

[7]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. Google Scholar

[8]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar

[9]

A. EdenC. FoiasB. Nicolaenko and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 559-562. Google Scholar

[10]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for Dissipative Evolution Equations, John Wiley & Sons, Chichester, 1994. Google Scholar

[11]

M. EfendievA. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X. Google Scholar

[12]

M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation, Discrete Cont. Dyn. Syst., 35 (2015), 2539-2564. doi: 10.3934/dcds.2015.35.2539. Google Scholar

[13]

A. Khanmamedov, Strongly damped wave equation with exponential nonlinearities, J. Math. Anal. Appl., 419 (2014), 663-687. doi: 10.1016/j.jmaa.2014.05.010. Google Scholar

[14]

P. KloedenJ. Real and C. Sun, Robust exponential attractors for non-autonomous equations with memory, Commun. Pure Appl. Anal., 10 (2011), 885-915. doi: 10.3934/cpaa.2011.10.885. Google Scholar

[15]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasgow Math. J., 46 (2004), 131-141. doi: 10.1017/S0017089503001605. Google Scholar

[16]

A. MiranvilleV. Pata and S. Zelik, Exponential attractors for singular perturbed damped wave equations: A simple construction, Asymptotic Anal., 53 (2007), 1-12. Google Scholar

[17]

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

[18]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. Google Scholar

[19]

F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I, Russian Journal of Mathematical Physics, 15 (2008), 301-315. doi: 10.1134/S1061920808030014. Google Scholar

[20]

A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptotic Anal., 87 (2014), 191-221. doi: 10.3233/ASY-131208. Google Scholar

[21]

M. Silva and T. Ma, Longtime dynamics for a class of Kirchhoff models with memory, J. Math. Phy. , 54 (2013), 021505, 15pp. doi: 10.1063/1.4792606. Google Scholar

[22] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[23]

Y. Wang and C. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Cont. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189. Google Scholar

[24]

Z. Yang, Longtime behavior of the Kirchhoff type equation with strong damping on $\mathbb{R}^N$, J. Differential Equations, 242 (2007), 269-286. doi: 10.1016/j.jde.2007.08.004. Google Scholar

[25]

Z. Yang and Y. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278. doi: 10.1016/j.jde.2010.09.024. Google Scholar

[26]

J. ZhangP. KloedenM. Yang and C. Zhong, Global exponential $κ$-dissipative semigroups and exponential attraction, Discrete Cont. Dyn. Syst., 37 (2017), 3487-3502. doi: 10.3934/dcds.2017148. Google Scholar

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