Exponential decay for a nonlinear model for electrical conduction in biological tissues

We study the asymptotic convergence to a periodic steady state of the solution of a nonlinear system of equations with periodic boundary data modeling electrical conduction in biological tissues, both in the microscopic and in the homogenized version. Such model keeps into account the resistive behavior of the intracellular and extracellular domains and also the capacitive/resistive behavior of the lipidic cellular membrane. The rate of convergence is analyzed and the systems of equations satisfied by the asymptotic limits are exhibited, when the resistive behavior of the membrane is described by a strictly monotone and coercive nonlinear function. The special case of homogeneous boundary conditions is also investigated.