Attractors for Fitzhugh-Nagumo lattice systems with almost periodic nonlinear parts (Q2029747)

The FitzHugh-Nagumo Lattice Dynamical System (LDS) appears in many applications in biology and circuit models. For deterministic FitzHugh-Nagumo autonomous and noautonomous LDSs, the existence of global attractors is investigated in [\textit{A. Y. Abdallah}, J. Appl. Math. 2005, No. 3, 273--288 (2005; Zbl 1128.37055); \textit{X.-J. Li} and \textit{D.-B. Wang}, J. Math. Anal. Appl. 325, No. 1, 141--156 (2007; Zbl 1104.37047); \textit{E. Van Vleck} and \textit{B. Wang}, Physica D 212, No. 3--4, 317--336 (2005; Zbl 1086.34047); \textit{B. Wang}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, No. 5, 1673--1685 (2007; Zbl 1142.34034)]. On the other hand, the existence of global random attractors for the corresponding stochastic systems is studied in [\textit{A. Gu} and \textit{Y. Li}, Commun. Nonlinear Sci. Numer. Simul. 19, No. 11, 3929--3937 (2014; Zbl 1470.34162); \textit{A. Gu} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 24, No. 10, Article ID 1450123, 9 p. (2014; Zbl 1302.34013); \textit{J. Huang}, Physica D 233, No. 2, 83--94 (2007; Zbl 1126.37048); \textit{Y. Wang} et al., J. Difference Equ. Appl. 14, No. 8, 799--817 (2008; Zbl 1143.37050); \textit{Z. Wang} and \textit{S. Zhou}, Taiwanese J. Math. 20, No. 3, 589--616 (2016; Zbl 1357.37088)]. Here the authors investigate the existence of a homogeneous global attractors for a new family of nonautonomous deterministic Fitzhugh-Nagumo LDSs with nonlinearities. The family of LDSs is written in abstract form, the well-posedness of the problem is established, a family of processes associated with this system is defined, and the existence of a uniform absorbing set and the continuity for this family of process is verified. The uniform estimates on the tails of solutions with respect to initial data from the uniform absorbing set and the translations of the time symbol are introduced. Such estimates are needed to obtain the asymptotic compactness of the solution semigroup, then by semigroup theory. The uniform global attractor for the family of considered processes is then studied.