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 PoissonBoltzmann type (PB n) equation with a small dielectric parameter 2 and nonlocal nonlinearity which takes into consideration of the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck (PNP) system. Under Robin type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviors of one dimensional solutions of PB n equations as the parameter approaches to zero. In particular, we show that in case of electro-neutrality, i.e., α = β, solutions of 1-D PB n equations have the similar asymptotic behavior as those of 1-D PB equations. However, as α 6= β (local non-electroneutrality), solutions of 1-D PB n equations may have blow-up behavior which can not be found in 1-D PB equations. Such a difference between 1-D PB and PB n equations can also be verified by numerical simulations.