Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity

Abstract

In this paper, we consider stationary solutions of the following one-dimensional Schnakenberg model with heterogeneity: \begin{document}$ \begin{equation*} \begin{cases} u_t-\varepsilon ^2 u_{xx} = d\varepsilon -u+g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ \varepsilon v_t-Dv_{xx} = \frac{1}{2}-\frac{c}{\varepsilon}g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ u_x (\pm 1) = v_x (\pm 1) = 0 . \end{cases} \end{equation*} $\end{document} We concentrate on the case that $ d, c, D>0 $ are given constants, $ g(x) $ is a given symmetric function, namely $ g(x) = g(-x) $, and $ \varepsilon>0 $ is sufficiently small and are interested in the effect of the heterogeneity $ g(x) $ on the stability. For the case $ g(x) = 1 $ and $ d = 0 $, Iron, Wei, and Winter (2004) studied the existence of $ N- $peaks symmetric stationary solutions and their stability. In this paper, first we construct symmetric one-peak stationary solutions $ (u_{\varepsilon}, v_{\varepsilon}) $ by using the contraction mapping principle. Furthermore, we give a linear stability analysis of the solutions $ (u_{\varepsilon}, v_{\varepsilon}) $ in details and reveal the effect of heterogeneity on the stability, which is a new phenomenon compared with the case $ g(x) = 1 $.