Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space

¹ Graduate School of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan
² Information Systems Department, Information & Communication Devision, Kyushu Electric Power Co. Inc., 810-8720 Fukuoka, Japan
³ Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan
yamamoto@math.tohoku.ac.jp
⁴ Faculty of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan
kawashim@math.kyushu-u.ac.jp

Received: 26 September 2011

Abstract

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝⁿ admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted L^p energy method.