  We derive thermodynamically consistent models of reaction-diffusion equations
coupled to a heat equation. While the total energy is conserved, the total
entropy serves as a driving functional such that the full coupled system is a
gradient flow. The novelty of the approach is the Onsager structure, which is
the dual form of a gradient system, and the formulation in terms of the
densities and the internal energy. In these variables it is possible to assume
that the entropy density is strictly concave such that there is a unique
maximizer (thermodynamical equilibrium) given linear constraints on the total
energy and suitable density constraints.
  We consider two particular systems of this type, namely, a diffusion-reaction
bipolar energy transport system, and a drift-diffusion-reaction energy
transport system with confining potential. We prove corresponding
entropy-entropy production inequalities with explicitely calculable constants
and establish the convergence to thermodynamical equilibrium, at first in
entropy and further in $L^1$ using Cziszar-Kullback-Pinsker type inequalities.
