Time Periodic Solutions For A Class Of Higher Order Parabolic Equations

Partial differential equations can reflect the restrictive relationship between the derivative of the unknown variables on the time and space variable, which can be used to depict a lot of mathematical models in many research fields. The parabolic equation is an important branch of partial differential equation,and the diffusion equation is a class of parabolic equation of great importance. It comes from the diffusion phenomenon, percolation theory, phase transition theory, biochemistry and biological groups, widely appeared in the nature, and other fields. Research on diffusion equation has always been closely concerned. Since the beginning of the19th century, mathematical work-ers started the research on diffusion equations. Of course, the research directions are not limited to the existence, uniqueness, regularity and the asymptotic behavior of the solution. So far, research on the second order parabolic equation is plenty, and has a relatively mature system and method, and many results have been extended to high dimensional case. Researchers have always paid much attention to the periodic prob-lems. In this paper, we will study the time periodic solution for a class of higher order parabolic equations.In the past few years, the periodic solution for the fourth-order population model and the Cahn-Hilliard equation has been fully researched. Y. P. Wang proved the exis-tence and uniqueness of the general time periodic solution for the periodic problem of the following equation in the one dimensional space. Y. L. Zhang proved the existence and uniqueness of the global weak time periodic so-lution and the classical time periodic solution for the above equation in the two dimen-sional space.The Cahn-Hilliard type equation can be used to describe competition and exclusion of the biology groups, population diffusion phenomenon affected by the environment and so on. J. X. Yin and others proved the existence and uniqueness of the time periodic solution for the following Cahn-Hilliard equation with periodic potentials and sources to the Dirichlet boundary value problem. L.Yin and others proved the existence of the time periodic solution of the following equation under the Dirichlet boundary value condition. In addition, Y. H. Li researched the above equation which is studied with the viscous term, i.e. periodic potential and periodic source of viscous Cahn-Hilliard type equation, and demonstrated the existence of periodic solution.Y H. Li considered the existence and attractiveness of the time periodic solution for the following Cahn-Hilliard equation with periodic gradient potentials and sources under the Dirichlet boundary value condition. Moreover, Y H. Li researched the above equation with the viscous term, that is periodic gradient potential and periodic source of viscous Cahn-Hilliard model equation, and demonstrated the existence of periodic solution.L. Fang considered the the existence of the time periodic solution for the viscous Cahn-Hilliard equation with variable mobility, the Cahn-Hilliard equation with constant mobility and the Cahn-Hilliard equation with variable mobility.Research methods and results of the fourth order parabolic equation allow us deeply inspired. In this thesis, we will introduce the time period problem for two kinds of sixth order parabolic equations.The second chapter discusses the sixth order parabolic equations in two dimen-sional space, i.e. the following equation where Q=Ω×(R+),Ω(?)R2is a bound domain with smooth boundary,γ>0, φ(s,t)=a(t)|S|p1s-b(t)s,p≥2,a(t),b(t) are periodic functions in C1|α(R|) functions with positive constant period ω,α∈(0,1),ω>0,f(x,t) is a nontrivial function in C3+α,α/6(Q) with positive period ω. we employ the Leray-Schauder fixed theorem to prove the existence of the time periodic solution for the following equation under the boundary value condition and the periodic condition u(x,0)=u(x,ω),x∈Ω.In the third chapter, we will study the following sixth order Cahn-Hilliard type equation where Q=Ω×(0,+∞),Ω(?)R2is a bound domain with smooth boundary. ψ/(u,t)=-a(t)u3+b(t)u, a(t) and b(t) are Holder continuous functions defined on R+with period ω, f(x,t) belongs to the space Cα,α/6(Q) for some α∈(0,1) with f(x,0)=f(x,ω).Such equation has a wealth of background and theoretical connotation. The re-search on the periodic solution for the sixth order parabolic equation is just beginning. We will consider the existence of periodic solutions for the above problem in the two dimensional space. The main difficulty is brought by the fourth-order diffusion and convection term. We need to obtain the Holder norm estimate of△u. By means of Cam-panato space and Schauder type estimate we obtain the estimate we need, and finally employ the Leray-Schauder fixed point theorem to get the existence of the periodic solution to the time periodic problem.