PNP Models with the Same Positively Charged Valence

A steady-state Poisson-Nernst-Planck model with n ion species is studied under the assumption that 1 n − positively charged ion species have the same valence and there is only one negatively charged ion species. By re-scaling, this model can be viewed as a standard singularly perturbed system. Based on the explicit formulae for the solutions of its limit slow system and singular perturbation methods, the existence of the solutions is analyzed.


Introduction
The Poisson-Nernst-Planck model is a well-known model of ion transport, which plays a crucial role in the study of many biological and physical problems, such as ion channels in cell membranes [1] [2] and semiconductor devices [3].
A steady-state Poisson-Nernst-Planck model [ where Φ is the electric potential; i c is the concentration for the ith ion species; i z is the valence; ( ) Q x is the permanent charge of the channel; ( ) i x µ is the electrochemical potential; ( ) h x is the area of the cross-section of the channel; i  is the flux density; i D is the diffusion coefficient; r ε is the relative dielectric coefficient; 0 ε is the vacuum permittivity; k is the Boltzmann constant; T is the absolute temperature; and e is the elementary charge. with 0 c is a constant.
Migration of charges for ionic flow through ion channels is often described mathematically by the Poisson-Nernst-Planck model (1.1), which can be viewed as a simplified version of the Maxwell-Boltzmann equations [7] [8] and the Langevin-Poisson equations [9] [10] by focusing on the key features of biological functions. Recently, the model (1.1) has been greatly studied [11]- [17]. In [18], the author obtained the existence and uniqueness of solutions for systems (1.1) and (1.2) under the assumption that ( ) 0 Q x = and 2 n = . In [19], the authors completely solved the existence and uniqueness of solutions for the boundary and (1.2) is provided in [21] based on the geometric singular perturbation theory [22] [23] [24]. In this paper, we intend to study the dynamics of the classical Poisson-Nernst-Planck model under the following hypotheses.
By re-scaling, The model (1) is reduced to a standard singularly perturbed system of the following ( ) ( with the boundary condition, for 1, , Actually, for the case 2 n = , system (1.4) and (1.5) corresponds to the equations studied in [20]. Additionally, due to the above hypothesis (H1), system (1.4) and (1.5) is a special case of the equations studied in [21], but in this special case, the explicit formulae for the solutions of its limit slow system can be obtained, which is crucial for the analysis of the existence of solutions for systems (1.4) and (1.5) in this paper by combining the technique of the geometric singular perturbation theory.   a a a n a n n n

Limiting Fast Orbits and Limiting Slow Orbits on [0, 1]
By letting 0 ε = in (2.6), we obtain the critical manifold

Limiting Slow Orbits on [0, a]
Now we identify the limiting slow orbits l Λ on the critical manifold l  . By using a rescaling 1 1 , .
n n n u p zc zc z c q , and its limiting slow system is For system (2.12), the critical manifold is It follows that the limiting slow system on l  is

G. J. Lin Journal of Applied Mathematics and Physics
For convenience, we denote , , , , , , , Proof. By system (2.13), it follows that Journal of Applied Mathematics and Physics (2.13) and using the variation of constants formula, the formulas for in the statement can be obtained. 
We will identify the limiting fast and limiting slow orbits which is normally hyperbolic.
a a a n z z z a a a n n n

Limiting Slow Orbits on [a, b]
Now we identify the limiting slow orbits m Λ on the critical manifold m  . By using a rescaling 1 1 , .

Limiting Fast Orbits on [b, 1]
In this section, we will identify the limiting fast and limiting slow orbits con-  which is normally hyperbolic.

Limiting Slow Orbits on [b, 1]
Now we identify the limiting slow orbits r Λ on the critical manifold r  . Just as in sections 2.1 and 2.2, it can be shown that the limiting slow system is       , , n J J −  are also determined. Therefore, all unknowns involved in Equations (3.27) and (3.28) are determined. Therefore, a limiting fast and limiting slow orbit is identified as follows, see