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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 3

### Mixed boundary conditions for a simplified quantum energy-transport model in multi-dimensional domains

Pages: 635 – 663

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n3.a4

#### Author

#### Abstract

In this paper we obtain a weak solution to a quantum energy-transport model for semiconductors. The model is formally derived from the quantum hydrodynamic model in the largetime and small-velocity regime by Jüngel and Milišič [Nonlinear Anal.: RealWorld Appl., 12:1033–1046, 2011]. It consists of a fourth-order nonlinear parabolic equation for the electron density, an elliptic equation for the electron temperature, and the Poisson equation for the electric potential. Our solution is global in the time variable, while the $N$ space variables lie in a bounded Lipschitz domain with a mixed boundary condition. The existence proof is based upon a carefully-constructed approximation scheme which generates a sequence of positive approximate solutions. These solutions are so regular that they can be used to form a variety of test functions to produce *a priori* estimates. Then these estimates are shown to be enough to justify passing to the limit in the approximate problems.

#### Keywords

Lipschitz domains, mixed boundary conditions, temperature gradient, degenerate fourth-order parabolic equations, quantum energy-transport model for semiconductors

#### 2010 Mathematics Subject Classification

35A01, 35B50, 35K52, 35K65, 35Q99, 74K35, 82D25