Time-periodic Problems And Initial-boundary Value Problems For Some Nonlinear Development Equations

In this paper,we study time-periodic problems and initial boundary value problems for nonlinear evolution equations.In chapterl,the research background and current situation of time-periodic problems and initial boundary value problems for nonlinear evolution equations are given.In Chapter2,the existence of the time-periodic solution to the time-dependent GinzburgLandau(TDGL)equations in the case of BCS-BEC crossover is established relying on the Leray-Schauder fixed point theorem.In Chapter3,we study the existence and uniqueness of time-periodic solutions to incompressible magnetohydrodynamic equations.Our approach combines energy estimates with topological degree theory,and our result follows from a limiting process.The main difficulty is to prove the compactness and continuity for a key operator Λ given in Definition 2.1,the proof is based on parabolic regularization.In Chapter4,we study the initial-boundary-value problem(IBVP)for coupled Kortewegde Vries equations posed on a finite interval with nonhomogeneous boundary conditions.We overcome the requirement for stronger smooth boundary conditions in the traditional method via the Laplace transform.Our approach uses the strong Kato smoothing property and the contraction mapping principle.