Pattern formation of a coupled two-cell Brusselator model

Abstract

In this paper, we study the stationary problems for the coupled two-cell Brusselator model as follows { − d 1 Δ u = 1 − ( b + 1 ) u + b u 2 v + c ( w − u ) in Ω , − d 2 Δ v = u − u 2 v in Ω , − d 1 Δ w = 1 − ( b + 1 ) w + b w 2 z + c ( u − w ) in Ω , − d 2 Δ z = w − w 2 z in Ω , ∂ ν u = ∂ ν v = ∂ ν w = ∂ ν z = 0 on ∂ Ω . We first study the stability of the unique positive constant solution ( 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 d 1 , d 2 and b varied.