Large Time Behavior of Solutions to n-Dimensional Bipolar Hydrodynamic Models for Semiconductors

In this paper, we study the n-dimensional ($n\geq1$) bipolar hydrodynamic model for semiconductors in the form of Euler–Poisson equations. In the 1-D case, when the difference between the initial electron mass and the initial hole mass is nonzero (switch-on case), the stability of nonlinear diffusion waves has been open for a long time. In order to overcome this difficulty, we ingeniously construct some new correction functions to delete the gaps between the original solutions and the diffusion waves in $L^2$-space, so that we can deal with the 1-D case for general perturbations, and prove the $L^\infty$-stability of diffusion waves in the 1-D case. The optimal convergence rates are also obtained. Furthermore, based on the 1-D results, we establish some crucial energy estimates and apply a new but key inequality to prove the stability of planar diffusion waves in n-D case. This is the first result for the multidimensional bipolar hydrodynamic model of semiconductors.