Exponential equilibration of an electro-energy-reaction-diffusion system arising from a two-level semiconductor model

We consider a thermodynamically correct framework for electro-energy-reaction-diffusion systems, which feature a monotone entropy functional while conserving the total charge and the total energy. For these systems, we construct a relative entropy functional, which acts as a Lyapunov functional, and we give an explicit expression for the related entropy production functional. The main result of this note is the derivation of a so-called entropy-entropy production inequality in a special situation, namely for a specific two-level semiconductor model of Shockley-Read-Hall type. Supposing that global weak solutions exist, we are able to prove that the relative entropy decays exponentially along trajectories of the underlying system. As a consequence, these global solutions converge to the equilibrium at an exponential rate.