Asymptotic stability of the stationary solution for a new mathematical model of charge transport in semiconductors

Authors:
A. M. Blokhin and D. L. Tkachev

Journal:
Quart. Appl. Math. **70** (2012), 357-382

MSC (2010):
Primary 35G61, 35D30; Secondary 82D37

DOI:
https://doi.org/10.1090/S0033-569X-2012-01251-7

Published electronically:
February 29, 2012

MathSciNet review:
2953108

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Abstract | References | Similar Articles | Additional Information

Abstract: We study an initial boundary value problem for a system of quasilinear equations which are effectively used for finding by the stabilization method numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). This initial boundary value problem has the following peculiarities: the PDE system is not a Cauchy-Kovalevskaya-type system; the boundary is a nonsmooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to a system of integro-differential equations. This allows one to prove the local-in-time existence and uniqueness of a weakened solution. Using a constructed energy integral and the Schauder fixed-point theorem, we prove the global-in-time solvability of the initial boundary value problem and justify the stabilization method under additional assumptions on the problem's data.

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Additional Information

**A. M. Blokhin**

Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia; Novosibirsk State University, Novosibirsk, 630090, Russia

Email:
blokhin@math.nsc.ru

**D. L. Tkachev**

Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia; Novosibirsk State University, Novosibirsk, 630090, Russia

Email:
tkachev@math.nsc.ru

DOI:
https://doi.org/10.1090/S0033-569X-2012-01251-7

Keywords:
Non-Cauchy-Kovalevskaya-type system,
weakened solution,
local- and global-in-time solvability,
asymptotic (Lyapunov’s) stability,
stabilization method

Received by editor(s):
October 13, 2010

Published electronically:
February 29, 2012

Additional Notes:
The authors are indebted to Yu. L. Trakhinin and S. A. Boyarsky for their help in the preparation of the manuscript of this paper. This work was financially supported by RFBR project, 10-01-00320-a, the interdisciplinary project of basic research SB RAS-2009 (No.91), and the project “Development of scientific potential of the Higher School” 2009-2010 (No. 2.1.1/4591).

Article copyright:
© Copyright 2012
Brown University

The copyright for this article reverts to public domain 28 years after publication.