Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension

We consider a class of one-dimensional reaction-diffusion systems, \[ \left\{ \begin{array} [ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right. \] with homogeneous Neumann boundary conditions on a one dimensional interval. Under some generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\rightarrow0,$ such a system admits a steady state for which $u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$ and for sufficiently small $\tau$, such a steady state is known to be stable in time. On the other hand, it is also known that in the so-called shadow limit $D\rightarrow\infty,$ patterns having more than one interface are unstable. In this paper we analyse in detail the transition between the stable patterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. We show that this transition occurs when $D$ is exponentially large in $\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\gg D_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if $D D_{K}$. We also study the dynamics of the interfaces when $D$ is exponentially large; this allows us to describe in detail the mechanism leading to the instability. Direct numerical computations of stability and dynamics are performed, and these results are in excellent agreement with corresponding results as predicted by the asymptotic theory.