Existence And Uniqueness Of Weak Solutions For A Class Of Fourth-order Parabolic Equation

In the past few decades,the fourth-order parabolic partial differential equations have drawn great interest of mathematicians both in theoretical research and practical application.Many authors have studied the fourth-order parabolic partial differential equations in depth,such as the existence,uniqueness,asymptotic behavior,regularity and so on.At present,it is mainly applied in the fields of materials science,biological science,engineering,image processing,etc.In this thesis,we establish a class of fourth-order initial-boundary value problem of a two-dimensional low-curvature equation(?) Where ? is a bounded open domain of R2 with smooth boundary ?,the cylinder?T=?×(0,T],the lateral surface?:=?x(0,T],T is a positive number.Assume that u0?H01(?),g?C1(R) and for every t?g,g(t) is an increasing function such that|g(t)|?C|t|?,?>1 Where C is a positive constant.Without loss of generality,we may assume that g(0)=0.We prove the existence and uniqueness of weak solutions of the initial-boundary value problem directly by employing the regularization,difference and variation methods and Trudinger-Moser inequality,and the asymptotic behavior of the weak solution is analyzed.First,according to the method of time discretization,the solution of the corresponding elliptic initial-boundary value problem is obtained.Next,the weak solution of parabolic initial-boundary value problem with viscous term is derived by induction and energy estimation.Finally,the existence of weak solutions for the fourth-order parabolic initial-boundary value problem discussed in this paper is proved by eliminating the viscous term.Furthermore,the uniqueness and the long-term asymptotic behavior of weak solution is considered.