Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues

We study here a problem arising in electrical conduction in biological tissues. From a physical point of view it consists in the study of the electrical currents crossing a living tissue when an electrical potential is applied at the boundary. Here the living tissue is regarded as a composite periodic domain made of extracellular and intracellular materials separated by a lipidic membrane which experiments prove to exhibit both conductive (due to ionic channels in the membrane) and capacitive behavior. We study the asymptotic convergence to a periodic steady state of the microscopic and the macroscopic electric potential for large times. We consider here the case where the conductive behavior of the cell membranes is nonlinear and monotone increasing but not coercive. In order to achieve our goal, we proceed via a Liapunov-style technique so that the rate of convergence is not quantified.