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

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*} $\end{document} We concentrate on the case that \begin{document}$ d, c, D>0 $\end{document} are given constants, \begin{document}$ g(x) $\end{document} is a given symmetric function, namely \begin{document}$ g(x) = g(-x) $\end{document} , and \begin{document}$ \varepsilon>0 $\end{document} is sufficiently small and are interested in the effect of the heterogeneity \begin{document}$ g(x) $\end{document} on the stability. For the case \begin{document}$ g(x) = 1 $\end{document} and \begin{document}$ d = 0 $\end{document} , Iron, Wei, and Winter (2004) studied the existence of \begin{document}$ N- $\end{document} peaks symmetric stationary solutions and their stability. In this paper, first we construct symmetric one-peak stationary solutions \begin{document}$ (u_{\varepsilon}, v_{\varepsilon}) $\end{document} by using the contraction mapping principle. Furthermore, we give a linear stability analysis of the solutions \begin{document}$ (u_{\varepsilon}, v_{\varepsilon}) $\end{document} in details and reveal the effect of heterogeneity on the stability, which is a new phenomenon compared with the case \begin{document}$ g(x) = 1 $\end{document} .