Global existence for long wave Hopf unstable spatially extended systems with a conservation law

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. This is a joint work with N. Gauss, G. Schneider and D. Zimmermann (Universität Stuttgart, Germany).