# Poisson-Nernst-Planck equations with steric effects - non-convexity and multiple stationary solutions

## TL;DR

This paper investigates the existence and stability of multiple stationary solutions in Poisson-Nernst-Planck equations with steric effects, revealing instability in the classical model and proposing a new stable, multi-solution framework.

## Contribution

It demonstrates the instability of multiple solutions in PNP-steric equations and introduces a novel PNP-Cahn-Hilliard model with stable, multi-stable solutions.

## Key findings

- Steric effects lead to multiple stationary solutions in PNP equations.
- Classical PNP-steric solutions are unstable under dynamics.
- The PNP-Cahn-Hilliard model admits stable, multi-stable solutions.

## Abstract

We study the existence and stability of stationary solutions of Poisson-Nernst- Planck equations with steric effects (PNP-steric equations) with two counter-charged species. These equations describe steady current through open ionic channels quite well. The current levels in open ionic channels are known to switch between `open' or `closed' states in a spontaneous stochastic process called gating, suggesting that their governing equations should give rise to multiple stationary solutions that enable such multi-stable behavior. We show that within a range of parameters, steric effects give rise to multiple stationary solutions that are smooth. These solutions, however, are all unstable under PNP-steric dynamics. Following these findings, we introduce a novel PNP-Cahn-Hilliard model, and show that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.