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A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis
On the forward dynamical behavior of nonautonomous systems
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 |
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.
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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