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With a small parameter $\epsilon$, Poisson-Nernst-Planck (PNP) systems over afinite one-dimensional (1D) spatial domain have steady state solutions, called1D boundary layer solutions, which profiles form boundary layers near boundarypoints and become at in the interior domain as $\epsilon$ approaches zero. Forthe stability of 1D boundary layer solutions to (time-dependent) PNP systems,we estimate the solution of the perturbed problem with globalelectroneutrality. We prove that the $H^{-1}_x$ norm of the solution of theperturbed problem decays exponentially (in time) with exponent independent of$\epsilon$ if the coefficient of the Robin boundary condition of electrostaticpotential has a suitable positive lower bound. The main difficulty is that thegradients 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 theperturbed problem into another parabolic system with a new and useful energylaw for the proof of the exponential decay estimate.

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