The search for the structure which can appear in singular limits of reaction-diffusion systems

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What differential equations can be obtained as singular limits of reaction-diffusion systems? In the singular limits of the reaction-diffusion systems some effects by diffusion are generally degenerate in some sense : in fact, some types of systems which exhibit spatial singularities like interfaces are known to be certain singular limits of appropriate reaction-diffusion systems. Can a differential equation which exhibit no singularities become a singular limit of an appropriate reaction-diffusion system? In this research we investigated whether some non-degenerate quasi-linear diffusion equations appearing in mathematical biology can be obtained as such limits. In particular, we focused on the quasi-linear diffusion equations called the Shigesada-Kawasaki-Teramoto model which describes spatial segregation phenomena for the population of two competitive species.We succeeded in constructing a reaction-diffusion system whose appropriate singular limit becomes the Shigesada-Kawasaki-Teramoto model. We proved the convergence of such a reaction-diffusion system to the Shigesada-Kawasaki-Teramoto model by an energy method and clarified the fact that the linear stability of a constant equilibrium in such a reaction-diffusion system coincides with that in the Shigesada-Kawasaki-Teramoto model. Such a reaction-diffusion system we constructed are also regarded as a reasonable model describing the mechanism of biological diffusion.Also we held a workshop on applied mathematics at Morioka three times to bring together information basic in this research. The results of the workshop are summarized in three proceedings which are delivered to more than one hundred researchers who investigate related topics.