| In this paper,we consider the complicated asymptotic behavior of the Cauchy problem of a class of Cahn-Hilliard equation,(?).As a type of typical higher-order parabolic partial differential equation,the Cahn-Hilliard equation is proposed when describing the phase separation and diffusion phenomenon caused by the sufficient cooling of two solutions.It is used in physics,biology,chemistry,etc.There are extremely wide applications in the field.This equation is based on a continuous model of phase transition in a binary system.In practice,we often see the long-time behavior of the system,which is manifested by the formation of patterns or the microstructure is effectively frozen into the system.The study of the complicated asymptotic behavior of higher-order parabolic equations is very necessary and has high theoretical value and good application prospects.The asymptotic behavior of the solution of the parabolic partial differential equation is to describe the nature of the solution when time approaches infinity,such as when the time is infinite,the solution converges to a fixed function or blow-up.The complicated asymptotic behavior is generally described by the number of elements in theωlimit set,and the number of elements must be at least two to represent a complicated asymptotic behavior.The complicated asymptotic behavior of solutions to second-order parabolic partial differential equations has attracted widespread interest,but little attention has been paid to the complicated asymptotic behavior of solutions to high-order parabolic partial differential equations.The main purpose of this article is to consider the complicated asymptotic behavior of the classical Cahn-Hilliard equation in the higher-order parabolic partial differential equation.We will use two different methods for research.One is to use the method of constructing the initial value to research;the other is to first establish the equivalent relationship between the solution and the initial value,and then consider the complicated asymptotic behavior of the solution.The full text is divided into four chapters.The first chapter is the introduction,which mainly introduces the current research status of the Cahn-Hilliard equation,specifically the general nonlinear terms Cahn-Hilliard equation,viscous Cahn-Hilliard equation,Cahn-Hilliard equation with gradient-dependent potential energy,and convection Cahn-Hilliard equation basic model,and introduces their existence and uniqueness,asymptotic behavior development status.Moreover,it introduces the research status of the complicated asymptotic behavior of various partial differential equations(Newton’s seepage equation,heat equation,Non-Newton’s seepage equation).The second chapter introduces the basic definitions and concepts of the prerequisite knowledge and the related inequalities.Chapter three,two different methods are used to discuss the complicated asymptotic behavior of the Cauchy problem of the Cahn-Hilliard equation.In the first method,the kernel of the evolution operator is first controlled appropriately by using the maximum kernel,and then the exchange relationship between the kernel of the evolution operator and the scale solution is established.Construct an initial value so that its scale solution limit set contains infinite elements,and use the number of elements of theωlimit set to prove that the solution of the Cauchy problem for the Cahn-Hilliard equation has complicated asymptotic behavior.It is proved that the solution of the Cauchy problem for the Cahn-Hilliard equation has complicated asymptotic behavior.For the second method,first find the asymptotic behavior of the scale solution in a certain norm and the asymptotic behavior of the initial value in a certain space,and then use the equivalent relationship to prove that the solutions of the Cahn-Hilliard equation to the Cauchy problem has complicated asymptotic behavior.It is revealed that complicated asymptotic behavior of higher order parabolic equations may occur.The fourth chapter is the conclusion and prospect of this article.The research methods and results of this paper are summarized,as well as the outlook for other nonlinear higher-order equations. |
