Existence and Local Uniqueness of Classical Poisson-Nernst-Planck Systems with Multi-Component Permanent Charges and Multiple Cations

We consider a quasi-one-dimensional Poisson-Nernst-Planck system for ionic flow through a membrane channel. Multi-component nonzero permanent charges, the major structural quantity of an ion channel, are included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for nonzero permanent charges is established. Numerical simulations are performed to roughly investigate the effects on ionic flows in terms of both I-V relations and individual fluxes from nonzero permanent charges.