Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model

The standard PNP model for ion transport in channels in cell membranes has been widely studied during the previous two decades; there is a substantial literature for both the dynamic and steady models. What is currently lacking is a generally accepted gating model, which is linked to the observed conformation changes on the protein molecule. In [SIAM J. Appl. Math. 61 (2000), no.3, 792{802], C.W. Gardner, the author, and R.S. Eisen-berg suggested a model for the net charge density in the infinite channel, which has connections to stochastic dynamical systems, and which predicted rectan-gular current pulses. The finite channel was analyzed by these authors in [J. Theoret. Biol. 219 (2002), no. 3, 291{299]. The finite channel cannot, in general, be analyzed by a traveling wave approach. In this paper, a rigorous study of the initial-boundary value problem is carried out for the determinis-tic version of the finite channel; an existence/uniqueness result, with a weak maximum principle, is derived on the space-time domain under assumptions on the inital and boundary data which confine the channel to certain states. Significant open problems remain and are discussed.