Osaka Journal of Mathematics

Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors

Shinya Nishibata and Masahiro Suzuki

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We study the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a one-dimensional hydrodynamic model of semiconductors. This problem is considered, in the previous researches [2] and [11], under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, this assumption is too narrow to cover the doping profile in actual diode devices. Thus, the main purpose of the present paper is to prove the asymptotic stability of the stationary solution without this assumption on the doping profile. Firstly, we prove the existence of the stationary solution. Secondly, the stability is shown by an elementary energy method, where the equation for an energy form plays an essential role.

Article information

Osaka J. Math., Volume 44, Number 3 (2007), 639-665.

First available in Project Euclid: 13 September 2007

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Zentralblatt MATH identifier

Primary: 82D37: Semiconductors 76E99: None of the above, but in this section 76N99: None of the above, but in this section
Secondary: 35L50: Initial-boundary value problems for first-order hyperbolic systems 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]


Nishibata, Shinya; Suzuki, Masahiro. Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors. Osaka J. Math. 44 (2007), no. 3, 639--665.

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  • K. Bløtekjær: Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices 17 (1970), 38--47.
  • P. Degond and P.A. Markowich: On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett. 3 (1990), 25--29.
  • Y. Deng, T.P. Liu, T. Yang and Z. Yao: Solutions of Euler-Poisson equations for gaseous stars, Arch. Ration. Mech. Anal. 164 (2002), 261--285.
  • C.L. Gardner: The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54 (1994), 409--427.
  • Y. Guo and W. Strauss: Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal. 179 (2006), 1--30.
  • D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983.
  • S. Kawashima, Y. Nikkuni and S. Nishibata: The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics; in Analysis of Systems of Conservation Laws (Aachen, 1997), Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 99, Chapman & Hall/CRC, Boca Raton, FL, 1999, 87--127.
  • S. Kawashima, Y. Nikkuni and S. Nishibata: Large-time behavior of solutions to hyperbolic-elliptic coupled systems, Arch. Ration. Mech. Anal. 170 (2003), 297--329.
  • S. Kawashima and S. Nishibata: Shock waves for a model system of the radiating gas, SIAM J. Math. Anal. 30 (1999), 95--117.
  • T. Luo and J. Smoller: Rotating fluids with self-gravitation in bounded domains, Arch. Ration. Mech. Anal. 173 (2004), 345--377.
  • H. Li, P. Markowich and M. Mei: Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 359--378.
  • T. Makino: Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars, Transport Theory Statist. Phys. 21 (1992), 615--624.
  • P.A. Markowich, C.A. Ringhofer and C. Schmeiser: Semiconductor Equations, Springer-Verlag, Vienna, 1990.
  • A. Matsumura and T. Murakami: Asymptotic behaviour of solutions of solutions for a fluid dynamical model of semiconductor equation, to appear.
  • R. Racke: Lectures on Nonlinear Evolution Equations, Friedr. Vieweg & Sohn, Braunschweig, 1992.
  • S. Schochet: The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), 49--75.