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Quantum Euler-Poisson systems: global existence and exponential decay. (English) Zbl 1069.35012

Summary: A one-dimensional transient quantum Euler-Poisson system for the electron density, the current density, and the electrostatic potential in bounded intervals is considered. The equations include the Bohm potential accounting for quantum mechanical effects and are of dispersive type. They are used, for instance, for the modelling of quantum semiconductor devices.
The existence of local-in-time solutions with small initial velocity is proven for general pressure-density functions. If a stability condition related to the subsonic condition for the classical Euler equations is imposed, the local solutions are proven to exist globally in time and tend to the corresponding steady-state solution exponentially fast as the time tends to infinity.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35Q60 PDEs in connection with optics and electromagnetic theory
35Q40 PDEs in connection with quantum mechanics
82D37 Statistical mechanics of semiconductors
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