  Electro-energy-reaction-diffusion systems are thermodynamically consistent
continuum models for reaction-diffusion processes that account for temperature
and electrostatic effects in a way that total charge and energy are conserved.
The question of the long-time asymptotic behavior in
electro-energy-reaction-diffusion systems and the characterization of their
equilibrium solutions leads to a maximization problem of the entropy on the
manifold of states with fixed values for the linear charge and the nonlinear
energy functional. As the main result, we establish the existence, uniqueness,
and regularity of solutions to this constrained optimization problem. Important
ingredients in the proof are tools from convex analysis and a reduced version
of the Lagrange functional. We also derive the time-dependent PDE system in the
framework of gradient systems, and we discuss the relations between stationary
states, critical points, and local equilibria.
