TURING PATTERNS IN GENERAL REACTION-DIFFUSION SYSTEMS OF BRUSSELATOR TYPE

We study the reaction-diffusion system { u(t) - d(1)Delta u = a - (b + 1) + f(u)v in Omega x (0, T), v(t) - d(2)Delta v = bu - f(u)v in Omega x (0, T), u(x, 0) = u(0)(x), v(x, 0) = v(0)(x) on Omega, partial derivative u/partial derivative v(x, t) = partial derivative u/partial derivative v(x, t) = 0 on partial derivative Omega x (0, T). Here Omega is a smooth and bounded domain in R-N (N >= 1), a, b, d(1), d(2) > 0 abd f is an element of C-1[0, infinity) is a non-decreasing function. The case f(u) = u(2) corresponds to the standard Brusselator model for autocatalytic oscillating chemical reactions. Our analysis points out the crucial role played by the nonlinearity f in the existence of Turing patterns. More precisely, we show that if f has a sublinear growth then no Turing patterns occur, while if f has a superlinear growth then existence of such patterns is strongly related to the inter-dependence between the parameters a, b and the diffusion coefficients d(1), d(2).