On a Nonlinear Parabolic Problem arising in the Quantum Diffusive Description of a Degenerate Fermion Gas
Résumé
This article studies, both theoretically and numerically, a nonlinear drift-diffusion equation describing a gas of fermions in the zero-temperature limit. The equation is considered on a bounded domain whose boundary is divided into an " insulating " part, where homogeneous Neumann conditions are imposed, and a " contact " part, where non-homogeneous Dirichlet data are assigned. The existence of stationary solutions for a suitable class of Dirichlet data is proven by assuming a simple domain configuration. The long-time behavior of the time-dependent solution, for more complex domain configurations, is investigated by means of numerical experiments.
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