The mathematical model of the transport and diffusion of ions in biological channels is considered. It is described by the three-dimensional nonlinear evolution classical Poisson–Nernst–Planck (cPNP) system of partial differential equations with nonlinear coupled boundary conditions. In particular the Chang–Jaffé (CJ) conditions are given on the input and output of a channel. The Robin boundary conditions on a potential are taken. Theorems on the existence, uniqueness and nonnegativity of local weak solutions, in the suitable Sobolev spaces, are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.

考虑了离子在生物通道中迁移和扩散的数学模型。它由具有非线性耦合边界条件的偏微分方程的三维非线性演化经典Poisson-Nernst-Planck（cPNP）系统描述。特别是在通道的输入和输出上给出了Chang-Jaffé（CJ）条件。取势上的Robin边界条件。证明了在合适的Sobolev空间中局部弱解的存在性，唯一性和非负性的定理。证明存在结果的主要工具是Schauder-Tychonoff不动点定理。