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Homogenization of degenerated reaction-diffusion equations

Sina Reichelt:

 

Many reaction-diff usion processes arising in civil engineering, biology, or chemistry take place in porous media, for instance concrete carbonation or the spread-out of substances in biological tissues. A porous medium can be modeled as a perforated domain, where the particles or holes are periodically distributed in the domain with the period length ε > 0.
Here the characteristic length scale of the microstructure, which is proportional to ε, e.g. the particles or holes, is much smaller compared to the overall size of the domain. Systems with such diff erences in the involved length scales are very difficult to handle numerically, because the step size of the algorithm has to be of order ε in order to resolve the qualitative behavior of the system. Therefore it is the aim to derive eff ective equations, independent of ε, which are ideally simpler and qualitatively describe the properties of the original system. In the classical case, the coefficients in the eff ective equation are homogeneous and in this sense, the passage ε → 0 is called homogenization.
In my talk I deal with a class of parabolic PDE, depending on ε, that allow for degenerated diff usion coefficients and nonlinear reaction terms. Using the method of two-scale convergence, I derive e ffective equations, which are de fined on a two-scale space. The two-scale space consists of the macroscopic domain and the microscopic unit cell attached to each point of the macroscopic domain.

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