The validity of phase diffusion equations and of Cahn–Hilliard equations for the modulation of pattern in reaction–diffusion systems

So-called phase diffusion equations and Cahn–Hilliard equations can be derived via multiple scaling analysis in order to describe slow modulations in time and space of stable and slightly unstable spatially periodic pattern. It is the purpose of this paper to explain the extent to which these formal approximations are valid by proving estimates between the formal approximations and true solutions of the original system. The proof is given for an abstract reaction–diffusion system as original system