  We study the asymptotic behavior of complex discrete evolution equations of
Ginzburg- Landau type. Depending on the nonlinearity and the data of the
problem, we find different dynamical behavior ranging from global existence of
solutions and global attractors, to blow up in finite time. We provide
estimates for the blow up time, depending not only on the initial data but also
on the size of the lattice. Some of the theoretical results, are tested by
numerical simulations.
