Very slow dynamics for some reaction-diffusion systems of the activator-inhibitor type

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We consider a bistable reaction-diffusion equation coupled with a time-dependent constrained condition $$\left\{ \begin{gathered} u_t = \varepsilon ^2 u_{xx} + \sigma ^{ - 2} [u(1 - u^2 ) - \xi ] \hfill \\ x \in I = (0,1), t > 0, \hfill \\ \xi _t = \int_I {udx} - \gamma \xi \hfill \\ \end{gathered} \right.$$ where γ, δ and e are positive constants. This equation lies in a framework of activator-inhibitor models which arise in biology. When e is sufficiently small, it is found that internal layers of widthO(e) appear in theu-component under the zero-flux boundary conditions, and that these layers propagate very slowly with velocityO(e−A/e) for some positive constantA.