  A class of energy-transport equations without electric field under mixed
Dirichlet-Neumann boundary conditions is analyzed. The system of degenerate and
strongly coupled parabolic equations for the particle density and temperature
arises in semiconductor device theory. The global-in-time existence of weak
nonnegative solutions is shown. The proof consists of a variable transformation
and a semi-discretization in time such that the discretized system becomes
elliptic and semilinear. Positive approximate solutions are obtained by
Stampacchia truncation arguments and a new cut-off test function.
Nonlogarithmic entropy inequalities yield gradient estimates which allow for
the limit of vanishing time step sizes. Exploiting the entropy inequality, the
long-time convergence of the weak solutions to the constant steady state is
proved. Because of the lack of appropriate convex Sobolev inequalities to
estimate the entropy dissipation, only an algebraic decay rate is obtained.
Numerical experiments indicate that the decay rate is typically exponential.
