New poisson-boltzmann type equations: One-dimensional solutions

The Poisson-Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. In this paper we study a new Poisson-Boltzmann type (PB-n) equation with a small dielectric parameter ε² and non-local nonlinearity which takes into consideration the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck system. Under Robintype boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB-n equations as the parameter ε approaches zero. In particular, we show that in case of electroneutrality, i.e. α = β, solutions of 1D PB-n equations have a similar asymptotic behaviour as those of 1D PB equations. However, as α ≠ β (nonelectroneutrality), solutions of 1D PB-n equations may have blow-up behaviour which cannot be found in 1D PB equations. Such a difference between 1D PB and PB n equations can also be verified by numerical simulations.