  A simplified transient energy-transport system for semiconductors subject to
mixed Dirichlet-Neumann boundary conditions is analyzed. The model is formally
derived from the non-isothermal hydrodynamic equations in a particular
vanishing momentum relaxation limit. It consists of a drift-diffusion-type
equation for the electron density, involving temperature gradients, a nonlinear
heat equation for the electron temperature, and the Poisson equation for the
electric potential. The global-in-time existence of bounded weak solutions is
proved. The proof is based on the Stampacchia truncation method and a careful
use of the temperature equation. Under some regularity assumptions on the
gradients of the variables, the uniqueness of solutions is shown. Finally,
numerical simulations for a ballistic diode in one space dimension illustrate
the behavior of the solutions.
