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: { − d 1 Δ u = a − ( b + 1 ) u + f ( u ) v + c ( w − u ) in Ω , − d 2 Δ v = b u − f ( u ) v in Ω , − d 1 Δ w = a − ( b + 1 ) w + f ( w ) z + c ( u − w ) in Ω , − d 2 Δ z = b w − f ( w ) z in Ω , ∂ ν u = ∂ ν v = ∂ ν w = ∂ ν z = 0 on ∂ Ω . Here Ω ⊂ R N ( N ⩾ 1 ) is a smooth and bounded domain, a , b , c , d 1 , d 2 are positive constants and f ∈ C 1 ( 0 , ∞ ) ∩ C [ 0 , ∞ ) is a nonnegative and nondecreasing function such that f > 0 in ( 0 , ∞ ) . When f ( u ) = u 2 , 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.