Asymptotic Stability of a Stationary Solution to a Thermal Hydrodynamic Model for Semiconductors

The present paper concerns the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a one-dimensional heat-conductive hydrodynamic model for semiconductors. It is important to analyze thermal influence on the motion of electrons in semiconductor device to improve the reliability of devices by handling a hot carrier problem. We show the unique existence of the stationary solution satisfying a subsonic condition by using the Leray–Schauder and the Schauder fixed-point theorems. Then the asymptotic stability of the stationary solution is proved by deriving the a priori estimate uniformly in time. Here an energy form plays an essential role. We also prove that the solution converges to the stationary solution exponentially fast as time tends to infinity.

[1]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[2]  Shock waves for a model system of the radiating gas , 1999 .

[3]  Pierre Degond,et al.  On a one-dimensional steady-state hydrodynamic model , 1990 .

[4]  Lawrence F. Shampine,et al.  Initial value problems , 2007, Scholarpedia.

[5]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[6]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[7]  Yan Guo,et al.  Stability of Semiconductor States with Insulating and Contact Boundary Conditions , 2006 .

[8]  Carl L. Gardner,et al.  The Quantum Hydrodynamic Model for Semiconductor Devices , 1994, SIAM J. Appl. Math..

[9]  Eberhard Zeidler,et al.  Linear monotone operators , 1990 .

[10]  R. Racke.  Lectures on nonlinear evolution equations , 1992 .

[11]  S. Schochet.  The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit , 1986 .

[12]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[13]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[14]  Yoshiko Nikkuni,et al.  Large-Time Behavior of Solutions to Hyperbolic-Elliptic Coupled Systems , 2003 .

[15]  Ansgar Jüngel,et al.  Quasi-hydrodynamic Semiconductor Equations , 2001 .

[16]  ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS TO THE FULL 1D HYDRODYNAMIC MODEL FOR SEMICONDUCTORS , 2002 .

[17]  Dario Bini,et al.  Global Existence and Relaxation Limit for Smooth Solutions to the Euler-Poisson Model for Semiconductors , 2000, SIAM J. Math. Anal..

[18]  C. Schmeiser,et al.  Semiconductor equations , 1990 .

[19]  Shinya Nishibata,et al.  Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors , 2007 .

[20]  K. Blotekjaer.  Transport equations for electrons in two-valley semiconductors , 1970 .

[21]  Ming Mei,et al.  Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[22]  Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits , 2008 .