zbMATH — the first resource for mathematics

Attractors of reaction diffusion systems on infinite lattices. (English) Zbl 1041.37040
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.

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
35B41 Attractors
35K57 Reaction-diffusion equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
Full Text: DOI