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

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.