On ring-like solutions for the Gray–Scott model: existence, instability and self-replicating rings

Abstract

We consider the following Gray–Scott model in $B_R (0) = \{x: |x|\lt R\} \subset \R^N, N=2,3$ : \[ \left\{ \begin{array}{@{}ll} v_t= \ep^2 \oldDelta v- v+ A v^2 u &\mbox{ in } B_R (0),\\ \tau u_t=\oldDelta u+1-u - v^2 u & \mbox{ in } B_R(0), \\ u, v\gt 0;~~ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu} =0 &\hspace*{3pt}\mbox{on} \ \partial B_R(0) \end{array} \right. \] where $\ep>0$ is a small parameter. We assume that $A= \skew{7}\hat{A} \ep^{\frac{1}{2}}$ . For each $\skew{7}\hat{A} and $R , we construct ring-like solutions which concentrate on an $(N-1)$ -dimensional sphere for the stationary system for all sufficiently small $\ep$ . More precisely, it is proved the above problem has a radially symmetric steady state solution $(v_{\ep,R}, u_{\ep, R})$ with the property that $v_{\ep,R} (r) \to 0 $ in $\R^N \backslash \{ r \not = r_0 \}$ for some $ r_0 \in (0, R)$ . Then we show that for $N=2$ such solutions are unstable with respect to angular fluctuations of the type $ \Phi (r) e^{ \sqrt{-1} m \theta}$ for some $m$ . A relation between $\skew{7}\hat{A}$ and the minimal mode $m$ is given. Similar results are also obtained when $\Om=\R^N$ or $\Om= B_{R_2} (0) \backslash B_{R_1} (0)$ or $\Om= \R^N \backslash B_R (0)$ .