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

Published online by Cambridge University Press:  16 January 2012

Ryo Kobayashi ,
Masakazu Yamamoto  and
Shuichi Kawashima
Ryo Kobayashi
Affiliation:
Graduate School of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan Information Systems Department, Information & Communication Devision, Kyushu Electric Power Co. Inc., 810-8720 Fukuoka, Japan
Masakazu Yamamoto
Affiliation:
Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan. [email protected]
Shuichi Kawashima
Affiliation:
Faculty of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan; [email protected]
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Abstract

We study the initial value problem for the drift-diffusion model arising in semiconductordevice simulation and plasma physics. We show that the corresponding stationary problem inthe whole space ℝn admits a unique stationary solution in ageneral situation. Moreover, it is proved that when n ≥ 3, a uniquesolution to the initial value problem exists globally in time and converges to thecorresponding stationary solution as time tends to infinity, provided that the amplitudeof the stationary solution and the initial perturbation are suitably small. Also, we showthe sharp decay estimate for the perturbation. The stability proof is based on the timeweighted Lp energy method.

Keywords

Drift-diffusion modelstabilitydecay estimatesweighted energy method
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 1097 - 1121
Copyright
© EDP Sciences, SMAI, 2012

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