# American Institute of Mathematical Sciences

June  2019, 24(6): 2657-2681. doi: 10.3934/dcdsb.2018269

## Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

 National Center for Theoretical Sciences, National Taiwan University, Taipei City 10617, Taiwan

* Corresponding author: Chia-Yu Hsieh

Received  January 2018 Revised  June 2018 Published  October 2018

In this paper, we consider radial solutions of the Poisson-Nernst-Planck (PNP) system with variable dielectric coefficients $\varepsilon g(x)$ in $N$-dimensional annular domains, $N≥2$. When the parameter $\varepsilon$ tends to zero, the PNP system admits a boundary layer solution as a steady state, which satisfies the charge conserving Poisson-Boltzmann (CCPB) equation. For the stability of the radial boundary layer solutions to the time-dependent radial PNP system, we estimate the radial solution of the perturbed problem with global electroneutrality. We generalize the argument of the one spatial dimension case (cf. [18]) and find a new way to transform the perturbed problem. By choosing a suitable weighted norm, we then derive the associated energy law which can be used to prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially in time with the exponent independent of $\varepsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound.

Citation: Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269
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