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

¹ Weierstraß-Institut für Angewandte Analysis und Stochastik, Anton-Wilhelm-Amo-Straße 39, 10117 Berlin, Germany
² Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstraße 36, 8010 Graz, Austria
³ Humboldt-Universität zu Berlin, Institut für Mathematik, Rudower Chaussee 25, 12489 Berlin, Germany

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 30 January 2025
Accepted: 1 February 2026

Abstract

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.