Exponential decay estimates for the stability of boundary layer solutions to Poisson-Nernst-Planck systems: one spatial dimension case

With a small parameter $\epsilon$, Poisson-Nernst-Planck (PNP) systems over a finite one-dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, which profiles form boundary layers near boundary points and become at in the interior domain as $\epsilon$ approaches zero. For the stability of 1D boundary layer solutions to (time-dependent) PNP systems, we estimate the solution of the perturbed problem with global electroneutrality. We prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially (in time) with exponent independent of $\epsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound. The main difficulty is that the gradients of 1D boundary layer solutions at boundary points may blow up as $\epsilon$ tends to zero. The main idea of our argument is to transform the perturbed problem into another parabolic system with a new and useful energy law for the proof of the exponential decay estimate.