Attractors of reaction diffusion systems on infinite lattices
The authors show that for a dynamical system that is generated by an implicit discretization of the system \(u_t= Du_{xx}+ f(u)\), where \(u(t,x)\in \mathbb{R}^k\), \(D\subset \mathbb{R}^{k\times k}\) and \(x\in\mathbb{R}\), that is \[ {1\over h} (u^{n+1}_m- u^n_m)= {D\over m^2} (u^{n+1}_{m+1}- 2u^{n+1}_{m+1}+ u^{n+1}_{m-1})+ f(u^{n+1}_m), \] \(m\in\mathbb{R}\), \(n\in\mathbb{N}\) there exists global attractor \(A_{gl}\). It is shown that the size of the absorbing set containing \(A_{gl}\), is uniformly bounded for small time and space steps. The authors also establish upper semicontinuity of \(A_{gl}\) for the scalar case. Finally, the authors present an example in which the global attractor is infinite dimensional. It is based on the fact, the unstable manifold of the origin and the latter one may be infinite dimensional in case of a bimodal nonlinearity.
scientific article; zbMATH DE number 2057860