Stability Of A Heat Equation In Non-cylindrical Domains

This paper mainly studies the stability of a type of heat equation containing the unstable term Ru in the non-cylindrical domain.Firstly,it is mainly judged whether the system is stable according to the trend of the system energy,and it is concluded that the system is unstable under certain conditions.Subsequently,internal control is applied to the unstable system to achieve exponential stability.Finally,examples are used to verify the accuracy of the main results obtained.This paper is divided into four chapter:The first chapter,the introduction part first introduces the history and research status of the subject,then introduces some symbols,definitions and known conclusions to be used in the article,and finally gives the main issues to be studied in this article.In Chapter 2,first,under certain assumptions,discuss the monotonicity and boundedness of the first eigenvalue of the operator-? in the time-varying domain,and its continuity.Then discuss the stability of the system through the method of energy estimation,and conclude that the stability of the system is related to the value of R.Secondly,internal control is applied to the unstable system and it is proved that it can achieve exponential stability.Finally,we also apply the equivalent transformation method to convert the stability problem of the heat equation in the time-varying domain into the stability problem of the parabolic equation with variable coefficients in the fixed domain to solve it.In Chapter 3,two examples of one-dimensional and two-dimensional numerical simulations are given to verify the accuracy of the conclusions.Chapter 4 summarizes the main results of this paper.Under certain assumptions,applying control to the unstable heat equation in the non-cylindrical domain in a fixed area inside can stabilize it.