Title: Poisson-Nernst-Planck equations with steric effects - non-convexity and multiple stationary 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.