  We are interested in reaction-diffusion systems, with a conservation law,
exhibiting a Hopf bifurcation at the spatial wave number $k = 0$. With the help
of a multiple scaling perturbation ansatz a Ginzburg-Landau equation coupled to
a scalar conservation law can be derived as an amplitude system for the
approximate description of the dynamics of the original reaction-diffusion
system near the first instability. We use the amplitude system to show the
global existence of all solutions starting in a small neighborhood of the
weakly unstable ground state for original systems posed on a large spatial
interval with periodic boundary conditions.
