STABILITY OF MULTI-PEAK SYMMETRIC STATIONARY SOLUTIONS FOR THE SCHNAKENBERG MODEL WITH PERIODIC HETEROGENEITY

In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity: {u(t) - epsilon(2)u(xx) = d epsilon - u + g(x)u(2)v, x is an element of (-1, 1), t > 0, epsilon v(t) - Dv(xx) = 1/2 - c/epsilon g(x)u(2)v, x is an element of (-1, 1), t > 0, u(x) (+/- 1) = v(x) (+/- 1) = 0. where d, c, D > 0 are given constants, epsilon > 0 is sufficiently small, and g(x) is a given positive function. Let N >= 1 be an arbitrary natural number. We assume that g(x) is a periodic and symmetric function, namely g(x) = g(-x) and g(x) = g(x + 2N(-1)). We study the stability of N-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity g(x) above on their stability. For the standard Schnakenberg model, namely the case of g(x) = 1, with d = 0, the stability of N-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of N-peak solution. In particular, we investigate how N-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case g(x) = 1.