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.