| Abstract: |
| Introduced independently by Fisher and Kolmogorov, Petrovski and Piskunov in 1937 to study the spatial propagation of a genetic trait, this type of equations has since been used in various contexts, ranging from population dynamics to ecology, biology, epidemiology, phase changes, combustion theory, and to social sciences and other fields. In this project, we want to extend the questions described above to the case where the diffusion is not necessarily homogenous or linear i.e. when the diffusion operator is either degenerate elliptic or singular elliptic, or it could be degenerate elliptic and fully non-linear. This is a very first attempt in this direction and a work in progress. In particular I will show some interesting new phenomena that arise when the operator is one of the so called truncated Laplacians that have been studied in recently in works in collaboration with Galise and Ishii. These would model diffusions that are only in the direction of some eigenvalues of the Hessian, i.e. the diffusion is in the directions where the growths is more decreasing or more increasing depending on the choice of the model. |
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