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Poisson-Nernst-Planck systems for ion channels with permanent charges. (English) Zbl 1137.34022

From the authors’ abstract: Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-Nernst-Planck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. The PNP system is studied for two types of ions with three regions of piecewise constant permament charge, assuming that the Debye number is large (because the electric field is very strong as compared to diffusion). Reservoirs are represented by the outer regions with permanent charge zero. If the reciprocal of the Debye number is viewed as a small parameter, the PNP system can be treated as a singularly perturbed system with two limiting (fast and slow) systems. A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. Existence and (local) uniqueness of the solution near each singular orbit is proved. A set of simultaneous equations appear in the construction of singular orbits. Multiple solutions of such equations might explain multiple valued phenomena which occur in biological channels.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A26 Geometric methods in ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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