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

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Abstract

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 ε are positive constants. This equation lies in a framework of activator-inhibitor models which arise in biology. When ε is sufficiently small, it is found that internal layers of widthO(ε) appear in theu-component under the zero-flux boundary conditions, and that these layers propagate very slowly with velocityO(e −A/ε) for some positive constantA.

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Authors and Affiliations

  1. Department of Mathematics, Hiroshima University, 724, Higashi-Hiroshima, Japan

    M. Kuwamura, S. -I. Ei & M. Mimura

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  1. M. Kuwamura
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  2. S. -I. Ei
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  3. M. Mimura
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Kuwamura, M., Ei, S.I. & Mimura, M. Very slow dynamics for some reaction-diffusion systems of the activator-inhibitor type. Japan J. Indust. Appl. Math. 9, 35–77 (1992). https://doi.org/10.1007/BF03167194

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  • DOI: https://doi.org/10.1007/BF03167194

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