Long-time Behavior of Solutions to the Bipolar Hydrodynamic Model of Semiconductors with Boundary Effect

Abstract

For a bipolar hydrodynamic model of semiconductors in the form of Euler–Poisson equations with Dirichlet or Neumann boundary conditions, in this paper we first heuristically analyze the most probable asymptotic profile (the so-called diffusion waves) and then prove this long-time behavior rigorously. For this, we construct correction functions to show the convergence of the original solution to the diffusion wave with optimal convergence rates by the energy method. Moreover, in the case with Dirichlet boundary condition, when the initial perturbation is in some weighted $L^1$ space, a faster and optimal convergence rate is also given.