On the equilibrium solutions of electro-energy–reaction–diffusion systems

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 of electro-energy–reaction–diffusion systems motivates the characterization of their equilibrium solutions, which leads to a maximization problem of the entropy on the manifold of states with fixed values for the linear charge and the nonlinear convex energy functional. As the main result, we establish the existence, uniqueness, and regularity of solutions to this constrained optimization problem. We give two conceptually different proofs, which are related to different perspectives on the constrained maximization problem. The first one is based on the method of Lagrange multipliers, while the second one employs the direct method of the calculus of variations.