A reaction-diffusion system of $\lambda$–$\omega$ type Part I: Mathematical analysis

We study two coupled reaction-diffusion equations of the $\lambda$–$\omega$ type [11] in $d\,{\le}\,3$ space dimensions, on a convex bounded domain with a $C^2$ boundary. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions [15] and compactness arguments. We also present a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations.