Pattern formation of a coupled two-cell Brusselator model

In this paper, we study the stationary problems for the coupled two-cell Brusselator model as follows{−d1Δu=1−(b+1)u+bu2v+c(w−u)inΩ,−d2Δv=u−u2vinΩ,−d1Δw=1−(b+1)w+bw2z+c(u−w)inΩ,−d2Δz=w−w2zinΩ,∂νu=∂νv=∂νw=∂νz=0on∂Ω. We first study the stability of the unique positive constant solution (u,v,w,z)=(1,1,1,1)(u,v,w,z)=(1,1,1,1). Then, we give a priori estimate (positive upper and lower bounds) to the positive solution. At last, we obtain the non-existence and existence of positive non-constant solutions as parameters d1d1, d2d2 and b varied.