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January  2020, 25(1): 473-487. doi: 10.3934/dcdsb.2019190

On the forward dynamical behavior of nonautonomous systems

Chunqiu Li 1, , Desheng Li 2, and  Xuewei Ju 3,,

1. 

College of Mathematical Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China
2. 

School of Mathematics, Tianjin University, Tianjin 300072, China
3. 

Department of Mathematics, School of Science, Civil Aviation University of China, Tianjin 300300, China

* Corresponding author: Xuewei Ju

Dedicated to Professor Peter E. Kloeden on the occasion of his 70th birthday

Received  September 2018 Revised  April 2019 Published  January 2020 Early access  September 2019

This paper is concerned with the forward dynamical behavior of nonautonomous systems. Under some general conditions, it is shown that in an arbitrary small neighborhood of a pullback attractor of a nonautonomous system, there exists a family of sets $ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $ of phase space $ X $, which is forward invariant such that $ \{\mathcal {A}_\varepsilon(p)\}_{p\in P} $ uniformly forward attracts each bounded subset of $ X $. Furthermore, we can also prove that $ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $ forward attracts each bounded set at an exponential rate.

Keywords: Nonautonomous dynamical systems, forward attraction, pullback attractors, exponential attraction.
Mathematics Subject Classification: 35B41, 37B25, 37B55, 37L30.
Citation: Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190
References:
[1]

A. Y. Abdallah, Uniformly exponential attractors for first order nonautonomous lattice dynamical systems, J. Diff. Equa., 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030.

[2]

A. V. Babin and B. Nicolaenko, Exponential attractor of reaction diffusion systems in an unbounded domain, J. Dyn. Diff. Equa., 7 (1995), 567-590.  doi: 10.1007/BF02218725.

[3]

T. Caraballo, J. A. Langa and R. Obaya, Pullback, forward and chaotic dynamics in 1D non-autonomous linear-dissipative equations, Nonlinearity, 30 (2017), 274-299.  doi: 10.1088/1361-6544/30/1/274.

[4]

A. Carvalho, J. A. Lange and J. Robinson, Attractors for Infinite-dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[5]

A. Carvalho, J. A. Lange, J. Robinson and A. Suárez, Characterization of nonautonomous attractors of a perturbed gradient system, J. Diff. Equa., 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.

[6]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[7]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[8]

D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144. 

[9]

V. V. Chepyzhov, Attractors of Mathematical Physics, Regional Conference Series in Mathematics 38, Amer. Math. Soc., Providence RI, 1978.

[10]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. 

[11]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equa., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[12]

R. Czaja, Pullback exponential attractors with adimissible exponential growth in the past, Nonlinear Anal., 104 (2014), 90-108.  doi: 10.1016/j.na.2014.03.020.

[13]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dyn. Diff. Equa., 13 (2001), 791-806.  doi: 10.1023/A:1016676027666.

[14]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37, Wiley, New York, 1994.

[15]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.

[16]

M. Efendiev, A. 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.

[17]

M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for nonautonomous dissipative system, J. Math. Soc. Japan, 63 (2011), 647-673. 

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.

[19]

X. W. Ju, D. S. Li and J. Q. Duan, Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations, Disc. Contin. Dyn. Syst. B, 24 (2019), 1175-1197. 

[20]

X. W. Ju, D.S. Li, C. Q. Li and A. L. Qi, Aproximate forward attractors of the nonautonomous dynamical systems, Chinese Annals of Mathematics, 40 (2019), 541-554. 

[21]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.

[22]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[23]

R. Temam, Infnite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[24] M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge University Press, Cambriage, England, 1992. 
[25]

Y. Wang, D. Li and P. E. Kloeden, On the asymptotical behavior of nonautonomous dynamical systems, Nonlinear Anal., 59 (2004), 35-53.  doi: 10.1016/j.na.2004.03.035.

[26]

Y. S. Zhong and C. K. Zhong, Exponential attractors for semigroups in Banach spaces, Nonlinear Anal.: Theory, Method & Applications, 75 (2012), 1799-1809.  doi: 10.1016/j.na.2011.09.020.

[27]

S. Zhou and X. Han, Pullback exponential attractors for nonautonomous lattice systems, J. Dyn. Diff. Equa., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

show all references

Dedicated to Professor Peter E. Kloeden on the occasion of his 70th birthday

References:
[1]

A. Y. Abdallah, Uniformly exponential attractors for first order nonautonomous lattice dynamical systems, J. Diff. Equa., 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030.

[2]

A. V. Babin and B. Nicolaenko, Exponential attractor of reaction diffusion systems in an unbounded domain, J. Dyn. Diff. Equa., 7 (1995), 567-590.  doi: 10.1007/BF02218725.

[3]

T. Caraballo, J. A. Langa and R. Obaya, Pullback, forward and chaotic dynamics in 1D non-autonomous linear-dissipative equations, Nonlinearity, 30 (2017), 274-299.  doi: 10.1088/1361-6544/30/1/274.

[4]

A. Carvalho, J. A. Lange and J. Robinson, Attractors for Infinite-dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[5]

A. Carvalho, J. A. Lange, J. Robinson and A. Suárez, Characterization of nonautonomous attractors of a perturbed gradient system, J. Diff. Equa., 236 (2007), 570-603.  doi: 10.1016/j.jde.2007.01.017.

[6]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[7]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[8]

D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144. 

[9]

V. V. Chepyzhov, Attractors of Mathematical Physics, Regional Conference Series in Mathematics 38, Amer. Math. Soc., Providence RI, 1978.

[10]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. 

[11]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equa., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[12]

R. Czaja, Pullback exponential attractors with adimissible exponential growth in the past, Nonlinear Anal., 104 (2014), 90-108.  doi: 10.1016/j.na.2014.03.020.

[13]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dyn. Diff. Equa., 13 (2001), 791-806.  doi: 10.1023/A:1016676027666.

[14]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37, Wiley, New York, 1994.

[15]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.

[16]

M. Efendiev, A. 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.

[17]

M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for nonautonomous dissipative system, J. Math. Soc. Japan, 63 (2011), 647-673. 

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.

[19]

X. W. Ju, D. S. Li and J. Q. Duan, Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations, Disc. Contin. Dyn. Syst. B, 24 (2019), 1175-1197. 

[20]

X. W. Ju, D.S. Li, C. Q. Li and A. L. Qi, Aproximate forward attractors of the nonautonomous dynamical systems, Chinese Annals of Mathematics, 40 (2019), 541-554. 

[21]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.

[22]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[23]

R. Temam, Infnite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[24] M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge University Press, Cambriage, England, 1992. 
[25]

Y. Wang, D. Li and P. E. Kloeden, On the asymptotical behavior of nonautonomous dynamical systems, Nonlinear Anal., 59 (2004), 35-53.  doi: 10.1016/j.na.2004.03.035.

[26]

Y. S. Zhong and C. K. Zhong, Exponential attractors for semigroups in Banach spaces, Nonlinear Anal.: Theory, Method & Applications, 75 (2012), 1799-1809.  doi: 10.1016/j.na.2011.09.020.

[27]

S. Zhou and X. Han, Pullback exponential attractors for nonautonomous lattice systems, J. Dyn. Diff. Equa., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

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