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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 Lp energy method.
@article{COCV_2012__18_4_1097_0,
author = {Kobayashi, Ryo and Yamamoto, Masakazu and Kawashima, Shuichi},
title = {Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1097--1121},
publisher = {EDP-Sciences},
volume = {18},
number = {4},
year = {2012},
doi = {10.1051/cocv/2011191},
mrnumber = {3019474},
zbl = {1262.35036},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011191/}
}
TY - JOUR AU - Kobayashi, Ryo AU - Yamamoto, Masakazu AU - Kawashima, Shuichi TI - Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 1097 EP - 1121 VL - 18 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011191/ DO - 10.1051/cocv/2011191 LA - en ID - COCV_2012__18_4_1097_0 ER -
%0 Journal Article %A Kobayashi, Ryo %A Yamamoto, Masakazu %A Kawashima, Shuichi %T Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 1097-1121 %V 18 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011191/ %R 10.1051/cocv/2011191 %G en %F COCV_2012__18_4_1097_0
Kobayashi, Ryo; Yamamoto, Masakazu; Kawashima, Shuichi. Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1097-1121. doi: 10.1051/cocv/2011191
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