## Abstract

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 ε^{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 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.

Original language | English (US) |
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Pages (from-to) | 431-458 |

Number of pages | 28 |

Journal | Nonlinearity |

Volume | 24 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2011 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics