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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.
[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.
[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.

[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.

[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.

[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.

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