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May  2022, 21(5): 1811-1831. doi: 10.3934/cpaa.2022048

Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay

1. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

2. 

Mathematisches Institut, Universität Tübingen, Tübingen 72076, Germany

* Corresponding author

Received  March 2021 Revised  December 2021 Published  May 2022 Early access  March 2022

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant No. 41875084

In this paper, we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.

Citation: Lin Yang, Yejuan Wang, Peter E. Kloeden. Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1811-1831. doi: 10.3934/cpaa.2022048
References:
[1]

M. Aouadi, Global and exponential attractors for extensible thermoelastic plate with time-varying delay, J. Differ. Equ., 269 (2020), 4079-4115.  doi: 10.1016/j.jde.2020.03.026.

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[3]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discret. Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.

[4]

Z. ChenB. Ermentrout and X. Wang, Wave propagation mediated by GABA B synapse and rebound excitation in an inhibitory network: a reduced model approach, J. Comput. Neuro., 5 (1998), 53-69. 

[5]

J. W. Cholewa and R. Czaja, Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors, J. Evol. Equ., 20 (2020), 485-515.  doi: 10.1007/s00028-019-00535-3.

[6]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differ. Equ., 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.

[7]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.

[8]

F. DongW. Li and L. Zhang, Entire solutions in a two-dimensional nonlocal lattice dynamical system, Commun. Pure Appl. Anal., 17 (2018), 2517-2545.  doi: 10.3934/cpaa.2018120.

[9]

A. De MasiT. Gobron and E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506.

[10]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, vol. 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[11]

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

[12]

M. Grasselli and D. Pra$\breve{z}$$\acute{a}$k, Exponential attractors for a class of reaction-diffusion problems with time delays, J. Evol. Equ., 7 (2007), 649-667.  doi: 10.1007/s00028-007-0326-7.

[13]

S. Habibi, Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.  doi: 10.2478/s12175-014-0272-0.

[14]

M. A. HammamiL. MchiriS. Netchaoui and S. Sonner, Pullback exponential attractors for differential equations with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 301-319.  doi: 10.3934/dcdsb.2019183.

[15]

X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467.  doi: 10.3934/dcds.2011.31.445.

[16]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[17]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013.

[18]

C. T. Lee, Non-local Concepts and Models in Biology, J. Theory Biol., 210 (2001), 201-219. 

[19]

S. MaP. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890.  doi: 10.1016/j.na.2005.10.042.

[20]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4612-0873-0.

[21]

E. Orlandi and L. Triolo, Travelling fronts in nonlocal models for phase separation in an external field, Proc. Roy. Soc. Edinb. Sect. A, 127 (1997), 823-835.  doi: 10.1017/S0308210500023854.

[22]

D. Pra$\breve{z}$$\acute{a}$k, Exponential attractors for abstract parabolic systems with bounded delay, Bull. Austral. Math. Soc., 76 (2007), 285-295.  doi: 10.1017/S0004972700039666.

[23]

Z. WangW. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.

[24]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010.

[25]

G. Zhang, Global stability of traveling wave fronts for non-local delayed lattice differential equations, Nonlinear Anal. Real World Appl., 13 (2012), 1790-1801.  doi: 10.1016/j.nonrwa.2011.12.010.

[26]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differ. Equ., 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.

show all references

References:
[1]

M. Aouadi, Global and exponential attractors for extensible thermoelastic plate with time-varying delay, J. Differ. Equ., 269 (2020), 4079-4115.  doi: 10.1016/j.jde.2020.03.026.

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[3]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discret. Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.

[4]

Z. ChenB. Ermentrout and X. Wang, Wave propagation mediated by GABA B synapse and rebound excitation in an inhibitory network: a reduced model approach, J. Comput. Neuro., 5 (1998), 53-69. 

[5]

J. W. Cholewa and R. Czaja, Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors, J. Evol. Equ., 20 (2020), 485-515.  doi: 10.1007/s00028-019-00535-3.

[6]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differ. Equ., 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.

[7]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.

[8]

F. DongW. Li and L. Zhang, Entire solutions in a two-dimensional nonlocal lattice dynamical system, Commun. Pure Appl. Anal., 17 (2018), 2517-2545.  doi: 10.3934/cpaa.2018120.

[9]

A. De MasiT. Gobron and E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506.

[10]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, vol. 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[11]

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

[12]

M. Grasselli and D. Pra$\breve{z}$$\acute{a}$k, Exponential attractors for a class of reaction-diffusion problems with time delays, J. Evol. Equ., 7 (2007), 649-667.  doi: 10.1007/s00028-007-0326-7.

[13]

S. Habibi, Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.  doi: 10.2478/s12175-014-0272-0.

[14]

M. A. HammamiL. MchiriS. Netchaoui and S. Sonner, Pullback exponential attractors for differential equations with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 301-319.  doi: 10.3934/dcdsb.2019183.

[15]

X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467.  doi: 10.3934/dcds.2011.31.445.

[16]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[17]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013.

[18]

C. T. Lee, Non-local Concepts and Models in Biology, J. Theory Biol., 210 (2001), 201-219. 

[19]

S. MaP. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890.  doi: 10.1016/j.na.2005.10.042.

[20]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4612-0873-0.

[21]

E. Orlandi and L. Triolo, Travelling fronts in nonlocal models for phase separation in an external field, Proc. Roy. Soc. Edinb. Sect. A, 127 (1997), 823-835.  doi: 10.1017/S0308210500023854.

[22]

D. Pra$\breve{z}$$\acute{a}$k, Exponential attractors for abstract parabolic systems with bounded delay, Bull. Austral. Math. Soc., 76 (2007), 285-295.  doi: 10.1017/S0004972700039666.

[23]

Z. WangW. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.

[24]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010.

[25]

G. Zhang, Global stability of traveling wave fronts for non-local delayed lattice differential equations, Nonlinear Anal. Real World Appl., 13 (2012), 1790-1801.  doi: 10.1016/j.nonrwa.2011.12.010.

[26]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differ. Equ., 224 (2006), 172-204.  doi: 10.1016/j.jde.2005.06.024.

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