This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.

本文涉及由离子传输网络引起的抛物线-椭圆-抛物线系统。它表明，对于任何适当规则的初始数据，与Neumann-Dirichlet边界条件相关的相应初始-边界值问题在一维设置中具有全局经典解，它均匀地有界并收敛于无穷大的稳态。具有时间衰减率或有限时间的时间。此外，通过采用问题的第三方程的零扩散极限，建立了其部分扩散对等点的整体弱解，并将完全扩散问题的解对部分扩散对等点的显式收敛速度，扩散率趋于零。