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# Homogenization of reaction-diffusion problems

Sina Reichelt:

Many mathematical models arising from biological, physical or technical problems involve effects on microscopic scales, e.g. spatial inhomogeneities of the underlying material. Those models depending on parameters or scales of the order ε challenge numerical and analytical treatment, for instance deriving local existence, independent of ε and ε-uniform bounds on solutions. Thus it is the subject of homogenization theory to find effective models defined on macroscopic scales, which approximate the properties of the original microscopic system well.

The subject matter are reaction-diffusion problems in divergence form with possibly nonlinear reactions terms, which are studied in several projects within the collaborative research center 910.

In the talk, basic methods of the homogenization of reaction-diffusion problems, where the spatial inhomogeneities admit an exact periodic structure in two different scales of the order ε and ε², will be presented. The question will be which are suitable convergences for the sequences of solutions to the inhomogeneous problems and how is the homogenized limit problem determined. The concepts of multi-scale convergence will give first answers to those questions.