On a drift-diffusion system for semiconductor devices (Q730133)

In the modeling of semiconductor devices, two nonlinear parabolic equations for the electron and hole densities are supplemented with an elliptic equation for the electrostatic potential (it states that a divergence of the related electrostatic potential depends on the difference of the relevant densities). A brief summary of the current state of art in the study of those Poisson-Nernst-Planck equations is given. In the present paper, the dissipation term in the parabolic equations is replaced by the fractional Laplacians (actually the Lévy generators) with different fractional powers. For such fractionally dissipative systems, the global existence of solutions is investigated and several decay estimates for the Lebesgue and Sobolev norms in one, two and three dimensions are established. The first term of the asymptotic expansion in the large-time limit is also provided.