Long Time Behavior Of Solutions Of Hamilton-Jacobi Equations With Time Periodic Hamiltonians

In this paper we study the long time behavior of viscosity solutions of time periodic Hamilton-Jacobi equations in the whole space Rn based on dynamical methods.In the preface we introduce the basic assumptions, the content of long time behavior of solutions of time periodic Hamilton-Jacobi equations, and the newest results.In chapter 1 we introduce some preliminary observations and some basic theorems.In chapter 2 we study the properties of time periodic Hamilton-Jacobi equa-tions. We give the existence result of viscosity solutions, and construct a periodic solution of the equation. Then we prove the existence of extremal curves of the periodic solutions.In chapter 3 we discuss the problems about the convergence of the solutions. First we give three convergence criteria of u(x, t+n) when the integer n goes to infinity. We prove a theorem of the viscosity solutions converges to the periodic solutions and obtain two representation formulas for the periodic asymptotic solutions.