In this thesis, we study 1-dimensional superconductivity with an applied electric current. The mathematical object of our work is a system of 1-dimensional partial differential equations, i.e., equations (1.18) and (1.19), derived from the simplest 3-dimensional time dependent Ginzburg-Landau equations (1.1) and (1.2). Our work focuses on the following three aspects, organized in three of the chapters.

First in chapter 2 we prove global existence of solution for the 1d TDGL (2.16) and (2.17). Under further assumptions, the existence of solution for the corresponding polar form equations (2.10) and (2.11) is proved. Issues about back flow of electric current are provided in the last section of this chapter.

Then in chapter 3 we study stability of normal state through a PT-symmetric eigenvalue problem (3.8) obtained from the linearization of the 1d TDGL (3.1) and (3.2). We then present our numerical result regarding the spectral structure of the eigenvalue problem (3.59) as the strength of the electric current varies. We introduce the interesting feature of eigenvalue collision and splitting in complex conjugate pairs.

Finally in chapter 4 we show our work on the asymptotic analysis for the system of time independent equations (4.6), (4.7), and (4.8) based on whether the ohmic conductivity σ is small or large. Some preliminary analytical results related to (4.6), (4.7), and (4.8) are also presented in this chapter.