Long Time Behavior Of Special Forms Of Schr(?)dinger Map

The Schr(?)dinger map solutions in quantum mechanics, differential geometry andother area are frontier topics and difficult problems. This paper studies the stability ofsolutions for a class of special form of Schr(?)dinger map. This thesis mainly containsthe following2aspects:In the first part we study the blow-up of2-dimensional heat flow of harmonicmap which can be seen as the dual form of the special form of Schr(?)dinger map. Inorder to research conveniently, we have to convert the form of equation but, at thesame time, it brings singularity existing in the equation. Anew comparison theoremneeds to be established. We further study the blow-up of2dimensional heat flowof harmonic map by using the new comparison principle.In the second part we generalize comparison theorem of the heat flow ofharmonic maps to special Schr(?)dinger map and study the long time behavior ofspecial Schr(?)dinger map. We investigate the existence of the upper and lowersolutions in the special Schr(?)dinger map model. Based on the above results, the factthat the solutions of special Schr(?)dinger map tend to stable solutions are obtained.In the last section, the dissertation is summarized. Moreover, the limitations andfurther studies on the solution for the high dimensional system are discussed.