一类非线性四阶抛物方程周期解的存在性

Existence of Time-Periodic Solutions to a Nonlinear Fourth-Order Parabolic Equation

研究了一类四阶抛物方程在一维情况下的时间周期解的存在性问题.方程形式上,最高阶为四阶线性微分项,低阶部分为二阶非线性微分项,赋予方程时间周期条件和边界条件.主要运用Galerkin方法构造基底及近似解,应用Leray-Schauder不动点定理得到该方程对应的线性方程解的存在性,利用近似解的一致性估计,并利用渐近极限的讨论,得到该方程时间周期解的存在性.

This paper 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 a 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.