The Existence Of Solutions For Several Higher-order Partial Differential Equations

This paper is devoted to studying the existence of time-periodic solutions of a nonlinear fourth-order parabolic equation,a viscous fourth-order parabolic equation and the existence of a thin film equation.For the method,the Galerkin method and Leray-Schauder fixed-point theorem are used to prove the existence of time-periodic solutions for the first two problems in chapter 2-3.In chapter 4,the entropy functional method is applied to prove the existence of weak solution for the thin film equation.In chapter 1,we introduce the background and recent development of fourth-order parabolic equations and we give the introduction of the main problems and results of this paper.Chapter 2 is devoted to studying the existence of time-periodic solutions of a fourth-order parabolic equation in one-dimensional space.Formally,the highest-order part is a fourth-order linear differential term,and the lower-order is second-order nonlinear differential term.Moreover,the time-periodic and boundary conditions are added for this problem.The Galerkin method is used to construct a base and the corresponding approximate solutions,and then the Leray-Schauder fixed-point theorem is applied to get the existence of the corresponding linear equation.By the uniform estimates of the approximate solutions and the argument of asymptotic limits,the existence of time-periodic solutions of this equation is obtained.In chapter 3,we give the existence of the time-periodic solutions of a fourth-order viscous parabolic equation.The Galerkin method is also used to construct the corresponding approximate solutions.Further,we make seme estimates to get the existence of the time-periodic solution of the equation.Chapter 4 is devoted to showing the existence of the weak solution of a fourth-order viscous parabolic equation in one-dimensional space.To overcome the difficulty from the nonlinear term,we construct approximate problem.The Galerkin method is used to prove the existence of approximate solution,and then the entropy functional method is used to yield the asymptotic limits.Finally,the existence of non-negative weak solution is proved by compactness arguments.