Pattern formation in a general two-cell Brusselator model

In this article, we are concerned with the following general coupled two-cell Brusselator-type system:{−d1Δu=a−(b+1)u+f(u)v+c(w−u)in Ω,−d2Δv=bu−f(u)vin Ω,−d1Δw=a−(b+1)w+f(w)z+c(u−w)in Ω,−d2Δz=bw−f(w)zin Ω,∂νu=∂νv=∂νw=∂νz=0on ∂Ω. Here Ω⊂RN(N⩾1) is a smooth and bounded domain, a,b,c,d1,d2 are positive constants and f∈C1(0,∞)∩C[0,∞) is a nonnegative and nondecreasing function such that f>0 in (0,∞). When f(u)=u2, this system corresponds to the coupled two-cell Brusselator model. In the present work, we exhibit the crucial role played by the nonlinearity f in generating the stationary patterns. Our conclusions show that if f has a sublinear growth then no stationary patterns occur, while if f has a superlinear growth, stationary patterns may exist.