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Large time behavior of the solutions to a hydrodynamic model for semiconductors. (English) Zbl 0936.35111
The authors consider the Cauchy problem for the one-dimensional isentropic Euler-Poisson model for semiconductors, where the energy equation is replaced by a pressure-density relation, which should have a positive derivative. They prove an existence and uniqueness result for the stationary solutions of the drift-diffusion equations, establish the global existence of smooth time-dependent solutions in a neighbourhood of this stationary solution, and prove the convergence to the stationary solution. Such a convergence is just a corollary from the exponentially decay estimates for the deviations in a small neighbourhood of the equilibrium.
A key argument of this work is that the given density of fixed positively charged background ions is stricktly positive and, hence, along with some additional assumptions, the unknown electron density function is positive. Problems arising at vacuum are beyond this paper.

MSC:
35L65 Hyperbolic conservation laws
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
78A35 Motion of charged particles
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