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), 357382
MSC (2010):
Primary 35G61, 35D30; Secondary 82D37
Published electronically:
February 29, 2012
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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 CauchyKovalevskayatype 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 integrodifferential equations. This allows one to prove the localintime existence and uniqueness of a weakened solution. Using a constructed energy integral and the Schauder fixedpoint theorem, we prove the globalintime 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:
http://dx.doi.org/10.1090/S0033569X2012012517
PII:
S 0033569X(2012)012517
Keywords:
NonCauchyKovalevskayatype system,
weakened solution,
local and globalintime 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, 100100320a, the interdisciplinary project of basic research SB RAS2009 (No.91), and the project “Development of scientific potential of the Higher School” 20092010 (No. 2.1.1/4591).
Article copyright:
© Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.
