Asymptotic Behaviour Of A Class Of Second Order Gradient System

Gradient system is a very important class of dynamic system, which has wide applications in mathematics, physics, engineering and some other fields. The study for gradient systems originates from the steepest decent problem. After many years research, the gradient system model becomes more and more complex. And researchers achieve many results in terms of stability and asymtotic behaviour of the solution of the gradient systems. Second-order gradient system is a standard class of oscillating model in physics. And its asymptotic behavior, which is the final state of the evolution system, has important researching value. In this thesis, we mainly discuss the global existence, uniqueness and asymptotic behavior of the solution of a class of second-order gradient system of multiple subsystems coupling with different damping.Firstly, we introduce the research background, significance and development process of the gradient system. Furthermore, we give brief description of the main contents of this thesis. Then, based on the theories of functional differential equations and nonlinear analysis, the global existence and uniqueness of the solution of the differential equation are studied. We establish Lyapunov function to study the basic properties of the solution of the gradient system. The convergence of the solution and the convergence rate are obtained by using ?ojasiewicz inequality for the analytic case. We also prove the convergence by studying the potential function for the convex and some other special cases. Finally, based on Yosida approximation theory, the global existence and properties of solution of the differential inclusion are acquired. And we also obtain the relationship between the second-order differential inclusions and the differential equations.