Problèmes de Di¤usion avec Interfaces et Applications Concrètes

Abstract

This work is devoted to the study of some dispersal problems describing a population dynamics in three habitats (including a refuge and two unfavorable ones) and which include individual behaviors at the borders between the regions. These problems are modeled by parabolic partial di¤erential equations. This kind of process is called skew Brownian motion. Taking into account the cylindrical geometry of patches or habitats, it is shown that the method that best adapts is that which uses the theory of di¤erential equations with operator coe¢ cients. The techniques used here are based essentially on the theory of semi-groups, the fractio- nal powers of linear operators, the interpolation spaces, and the H1- calculus for sectoral operators. In a rst step : the study of the existence, the unicity and the maximum regularity of the solution of the stationary linear problem allows us to obtain necessary and su¢ cient conditions on the data at the level of the interfaces. The second step is devoted to the study of the C0-semi-group analycity generated by the dispersion process in two habitats with a continuous dispersal condition at the interface. In the last step we make the spectral study of the initial problem which is very important for the analysis of the complete problem of evolution.