Upper Semicontinuity of the Attractor for Lattice Dynamical Systems of Partly Dissipative Reaction-diffusion Systems

Lattice dynamical systems arise in various application fields, for instance, in chemical reaction theory, material science, biology, laser systems, image processing and pattern recognition, and electrical engineering (cf. [6, 7, 13]). In each field, they have their own forms, but in some other cases, they appear as spatial discretizations of partial differential equations (PDEs). Recently, many authors studied various properties of the solutions for several lattice dynamical systems. For instance, the chaotic properties have been investigated in [1, 7, 8, 10, 11, 17], and the travelling solutions have been carefully studied in [2, 3, 7, 8, 9, 22]. From [18], we know that it is difficult to estimate the attractor of the solution semiflow generated by the initial value problem of dissipative PDEs on unbounded domains because, in general, it is infinite dimensional. Therefore, it is significant to study the lattice dynamical systems corresponding to the initial value problem of PDEs on unbounded domains because of the importance of such systems and they can be regarded as an approximation to the corresponding continuous PDEs if they arise as spatial discretizations of PDEs. The main idea of this work is originated from [4, 16]. In [4, 19, 20], the researchers proved the existence of global attractors for different lattice dynamical systems and they investigated the finite-dimensional approximations of these global attractors. In fact, Bates et al. [4] studied first-order lattice dynamical systems, and Zhou [20] gave a generalization of the result given by [4]. Zhou [19] studied a second-order lattice dynamical system and investigated the upper semicontinuity of the global attractor.