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 ² 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 Robin-type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB_n equations as the parameter 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 (non-electroneutrality), 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.