Fractional diffusion: biological models and nonlinear problems driven by the s-power of the Laplacian.

In the classical theory, the fractional diffusion is ruled by two different types of fractional Laplacians. Formerly known since 60s, the spectral fractional Laplacian had an important development in the recent mathematical study with the initial contributes of L. Caffarelli, L. Silvestre and X. Cabré, X.Tan. The integral version of the fractional Laplacian, recently discussed by M. Fukushima, Y. Oshima, M Takeda, and Song, Vondracek, is considered in a semilinear elliptic problem in presence of a general logistic function and an indefinite weight. In particular we look for a multiplicity result for the associated Dirichlet problem. In the second part, starting from the classical works of T.Hillen and G. Othmer and taking the Generalized velocity jump processes presented in a recent work of J.T.King, we obtain the fractional diffusion as limit of this last processes using the technique used in another recent work of Mellet, without the classical Hilbert or Cattaneo approximation's methods.