Abstract
A lot of processes coming from Physics, Chemistry, Biology, Economy, and other sciences can be described using systems of reaction-diffusion equations. In this chapter, we study the asymptotic behavior of the solutions of a system of infinite ordinary differential equations (a lattice dynamical system) obtained after the spacial discretization of a system of reaction-diffusion equations in an unbounded domain. This kind of dynamical systems is then of importance in the numerical approximations of physical problems.
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Abdallah AY (2005) Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete Contin Dyn Syst. doi:10.3934/dcdsb.2005.5.899
Ball JM (2000) Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. In Mechanics: from Theory to Computation. Springer, New York, pp 447–474
Bates PW, Lu K, Wang B (2001) Attractors for lattice dynamical systems. Int J Bifurcat Chaos. doi:10.1142/S0218127401002031
Bell J (1981) Some threshold results for models of myelinated nerves. Math Biosci 54:181–190
Beyn WJ, Pilyugin SYu (2003) Attractors of reaction diffusion systems on infinite lattices. J Dynam Differ Equat. doi:10.1023/B:JODY.0000009745.41889.30
Cellina A (1971) On the existence of solutions of ordinary differential equations in Banach spaces. Funkcialaj Ekvacioj 14:129–136
Chepyzhov VV, Vishik MI (2002) Attractors for equations of mathematical physics. American Mathematical Society, Providence
Cheskidov A, Foias C (2007) On global attractors of the 3D Navier-Stokes equations. J Differ Equat. doi:10.1016/j.jde.2006.08.021
Chow SN, Mallet-Paret J (1995) Pattern formation and spatial chaos in lattice dynamical systems. IEEE Trans Circ Syst. doi:10.1.1.45.8359
Chua LO, Yang Y (1998) Cellular neural networks: applications. IEEE Trans Circ Sys 1:1273–1290
Dunford N, Schwartz JT (1998) Linear operators, Part I. Wiley, NewYork
Erneux T, Nicolis G (1993) Propagating waves in discrete bistable reaction diffusion systems. Phys D. doi:10.1016/0167-2789(93)90208-I
Godunov AN (1975) Peano’s theorem in Banach spaces. Funct Anal Appl. doi:10.1007/BF01078180
Kapral R (1991) Discrete models for chemically reacting systems. J Math Chem. doi:10.1007/BF01192578
Kapustyan AV, Melnik VS, Valero J (2007) A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete Contin Dyn Syst. doi:10.3934/dcds.2007.18.449
Kapustyan AV, Valero J (2006) On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems. J Math Anal Appl. doi:10.1016/j.jmaa.2005.10.042
Kapustyan AV, Valero J (2007) Weak and strong attractors for the 3D Navier-Stokes system. J Differ Equat. doi:10.1016/j.jde.2007.06.008
Karachalios NI, Yannacopoulos AN (2005) Global existence and compact attractors for the discrete nonlinear Schrödinger equations. J Differ Equat. doi:10.1016/j.jde.2005.06.002
Kato S (1976) On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces. Funkcialaj Ekvacioj 19:239–245
Li X, Wang D (2007) Attractors for partly dissipative lattice dynamic systems in weighted spaces. J Math Anal Appl. doi:10.1016/j.jmaa.2006.01.054
Li X, Zhong Ch (2005) Attractors for partly dissipative lattice dynamic systems in l 2 ×l 2. J Comput Appl Math. doi:10.1016/j.cam.2004.09.014
Morillas F, Valero J (2005) Attractors for reaction-diffusion equations in R N with continuous nonlinearity. Asymptot Anal 44:111–130
Morillas F, Valero J (2009) A Peano’s theorem and attractors for lattice dynamical systems. Int J Bifurcat Chaos 19:557–578
Rosa R (2006) Asymptotic regularity conditions for the strong convergence towards weak limit set and weak attractors of the 3D Navier-Stokes equations. J Differ Equat. doi:10.1016/j.jde.2006.03.004
Sell G (1996) Global attractors for the three-dimensional Navier-Stokes equations. J Dynam Differ Equat. doi:10.1007/BF02218613
Swierniak A (1976) Generalization of Peano’s and Osgood’s theorems on differential equations in Banach spaces. Zesz Nauk Politech Slak 560:263–264
Van Vleck E, Wang B (2005) Attractors for lattice Fitz-Hugh-Nagumo systems. Phys D. doi:10.1016/j.physd.2005.10.006
Wang B (2006) Dynamics of systems on infinite lattices. J Differ Equat. doi:10.1016/j.jde.2005.01.003
Wang B (2007) Asymptotic behavior of non-autonomous lattice systems. J Math Anal Appl. doi:10.1016/j.jmaa.2006.08.070
Zhao C, Zhou Sh (2007) Limiting behaviour of a global attractor for lattice nonclassical parabolic equations. Appl Math Lett. doi:10.1016/j.aml.2006.06.019
Zhou S (2002) Attractors for lattice systems corresponding to evolution equations. Nonlinearity. doi:10.1088/0951-7715/15/4/307
Zhou S (2003) Attractors for first order dissipative lattice dynamical systems. Phys D. doi:10.1016/S0167-2789(02)00807-2
Zhou S (2004) Attractors and approximations for lattice dynamical systems. J Differ Equat. doi:10.1016/j.jde.2004.02.005
Zhou S, Shi W (2006) Attractors and dimension of dissipative lattice systems. J Differ Equat. doi:10.1016/j.jde.2005.06.024
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Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V. (2012). Attractors for Lattice Dynamical Systems. In: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Advances in Mechanics and Mathematics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28512-7_3
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