Justification of the stabilization method for a mathematical model of charge transport in semiconductors

An initial boundary value problem for a quasilinear system of equations is studied and effectively applied to numerically determine, by the stabilization method, stationary solutions of a hydrodynamic model describing the motion of electrons in the silicon transistor MESFET (metal semiconductor field effect transistor).
An initial boundary value problem has a number of special features; namely, the system of differential equations is not a system of Cauchy–Kovalevskaya type; the boundary of the domain is a nonsmooth curve, it contains corner points; the quasilinearity of the system is related, in particular, to the presence in the equations of squared component of the gradients of unknowns functions.
The problem under consideration can be reduced to an equivalent system of integrodifferential equations by using a representation of solutions of a model problem, which makes it possible to prove the local-in-time existence and uniqueness of a weakened solution.
Under additional assumptions on the problem data, the global solvability of the mixed problem is proved and the stabilization method is justified by using an energy integral constructed for this purpose and Schauder’s fixed point theorem.