  We investigate a class of anharmonic crystals in $d$ dimensions, $d\ge 1$,
coupled to both external and internal heat baths of the Ornstein-Uhlenbeck
type. The external heat baths, applied at the boundaries in the 1-direction,
are at specified, unequal, temperatures $\tlb$ and $\trb$. The temperatures of
the internal baths are determined in a self-consistent way by the requirement
that there be no net energy exchange with the system in the non-equilibrium
stationary state (NESS). We prove the existence of such a stationary
self-consistent profile of temperatures for a finite system and show it
minimizes the entropy production to leading order in $(\tlb -\trb)$. In the
NESS the heat conductivity $\kappa$ is defined as the heat flux per unit area
divided by the length of the system and $(\tlb -\trb)$. In the limit when the
temperatures of the external reservoirs goes to the same temperature $T$,
$\kappa(T)$ is given by the Green-Kubo formula, evaluated in an equilibrium
system coupled to reservoirs all having the temperature $T$. This $\kappa(T)$
remains bounded as the size of the system goes to infinity. We also show that
the corresponding infinite system Green-Kubo formula yields a finite result.
Stronger results are obtained under the assumption that the self-consistent
profile remains bounded.
