泊松─能斯特─普朗克類型方程之研究

Abstract

This thesis consists of three parts. The first part is devoted to the stability problem of boundary layer solutions to Poisson-Nernst-Planck (PNP) systems. PNP system has been widely used to describe the electron transport of semiconductors and the ion transport of ionic solutions, and plays a crucial role in the study of many physical and biological problems. PNP system with Robin boundary condition for the electrostatic potential admits a boundary layer solution as a steady state. We obtain some stability results by studying the asymptotic behavior for the steady state. In the second part, we study existence of solutions for a modified PNP system. Historically, from the Debye-Huckel theory, PNP systems are adopted for dilute ionic solutions. However, in biological system, ionic solutions are usually highly concentrated. The modified PNP system, which takes into account the relative drag from interaction of different ions, involves much more complicated nonlinear coupling between unknown variables. Compare with the original PNP system, it brings extra difficulties in analysis. We use Galerkin's method and Schauder's fixed-point theorem, and give an estimate of upper bounds of solutions to prove existence of local solutions. The third part deals with existence of solutions of a modified PNP equation with cross-diffusion. By using Galerkin's method and Schauder's fixed-point theorem, we obtain the local existence for this system. Moreover, we obtain the global existence by uniform in time L^2 estimates of solutions. We also consider a modified Keller-Segel system with similar modification and develop some local and global existence results.