Diffusive stability of spatially periodic solutions of the Brusselator   model
  Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg Landau amplitude equations, rigorously validating the standard weakly unstable approximation and Eckhaus criterion. 
Alim Sukhtayev, Kevin Zumbrun, Soyeun Jung, and Raghavendra   Venkatraman
math.AP