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Well-posedness of a nonlinear, nonlocal problem arising in ion transport
Abstract
We consider the well-posedness of two models for 1-dimensional movement of oxidized and reduced ions in a medium under the influence of an electric field. In the first model, we examine movement in both finite and semi-infinite length mediums. In the second model, the medium has finite length. There is an electrode located at one boundary. We perturb the potential at the electrode causing some of the oxidized ions near the electrode to gain electrons (becoming reduced). These concentrations changes at the electrode set up a two-way diffusion gradient: reduced ions move out into the bulk and oxidized ions move toward the electrode. The ions also migrate under the influence of an electric field. Our models for ion transport in this situation give rise to nonlinear parabolic PDE's for the charge concentrations. The equations in both models involve nonlocal source terms. In the first model, the nonlocal term involves a time derivative of the charge concentration, a complicating feature in the problem. For this model we establish the existence and uniqueness of a classical short-time solution, and various physical properties of the solution, like boundedness, spatial monotonicity, and decay as [special characters omitted] (when the medium is unbounded). Using analysis of the steady-state problem on a bounded interval, and numerical computation, we speculate on the qualitative behavior of these solutions. For the second model, we prove global existence of a unique weak solution.
Subject Area
Mathematics
Recommended Citation
Wagstrom, Rikki Bryanna Thompson, "Well-posedness of a nonlinear, nonlocal problem arising in ion transport" (1999). ETD collection for University of Nebraska-Lincoln. AAI9942163.
https://digitalcommons.unl.edu/dissertations/AAI9942163