Long-time Dynamics Of Solution For A Class Of Piezoelectric Beam Equations

Piezoelectric material is a class of smart materials that can convert mechanical energy and electrical energy into each other,and are scalable,smaller,and more efficient.This makes piezoelectric materials widely used in industry,biomedicine and other fields.The piezoelectric beam is one type of elastic beams with electrodes on the top and bottom surfaces,insulated edges,and connected to external circuits.The related research also has many values and significance.This paper mainly studies the long-time dynamics of piezoelectric beam with memory terms and time-varying delay terms,which mainly includes the existence of global attractor with finite fractal dimension and generalized exponential attractor by establishing quasi-stable estimate on the basis of the existence of global solution.In addition,the upper semi-continuity of global attractor and the energy decay estimate of the system are also established.The structure of this paper is as follows:In chapter 1,we introduce the research background and current research status of the piezoelectric beam at home and abroad,and give the main research contents and structure of this paper,as well as the introduction of new variables used to deal with memory term and time-varying delay term.In chapter 2,we present some concepts of semigroup theory and long-time dynamics,and the inequalities are needed below.In Chapter 3,the problem of the solution of the piezoelectric beam with memory term and time-varying delay term is studied.Based on the theory of semigroup combined with the energy inequality,the existence and properties of the global solution are proved.In Chapter 4,the long-time dynamics of the piezoelectric beam with memory term and time-varying delay term is considered.The existence of global attractor with finite fractal dimension is derived by quasi-stable estimate and generalized exponential attractor is obtained.Moreover,the upper semi-continuity of all ε ∈[0,1] is derived by proving the continuity of global attractor on a ‘residual’ dense set belongs to [0,1].In Chapter 5,we study that the exponential decay of energy of the piezoelectric beam with memory term and time-varying delay term.