指導教授：林太家臺灣大學：數學研究所謝佳佑Hsieh, Chia-YuChia-YuHsieh2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264009This 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.口試委員會審定書 i 誌謝 ii 中文摘要 iii Abstract iv Contents vi 1 Stability of Boundary Layer Solutions to Poisson-Nernst-Planck Systems 1 1.1 Introduction 1 1.2 Linearized Problem 9 1.2.1 Proof of Theorem 1.1.2 14 1.2.2 Corollaries of Theorem 1.1.2 20 1.3 Asymptotic Limit Equation of ( ilde{delta} , ilde{eta} ) 22 1.4 Stability for PNP system 25 1.4.1 Proof of Theorem 1.4.1 30 1.4.2 Corollaries of Theorem 1.4.1 32 2 Local Existence for Modified Poisson-Nernst-Planck Systems with Saturable Nonlinearity 34 2.1 Introduction 34 2.2 Local Existence 37 3 Global Existence for Drift-Diffusion Systems with Cross-Diffusion 62 3.1 Introduction 62 3.2 Local Existence 64 3.3 Global Existence 85 Bibliography 924454558 bytesapplication/pdf論文公開時間：2016/08/17論文使用權限：同意有償授權(權利金給回饋學校)泊松─能斯特─普朗克方程穩定性存在性凱勒─謝格爾方程thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264009/1/ntu-103-F96221005-1.pdf