The Large-time Behavior Of Solutions For Hamilton-Jacobi Equations

Hamilton-Jacobi equations arise from the variational method, is an important class of first order nonlinear partial differential equations, which are widely used.in classical mechanics, geometric optics, optimal control, differential games and so on..The study of the large-time behavior of solutions is an important aspect of the study for Hamilton-Jacobi equations, which is closely related with the problems of Hamiltonian dynamics, physics and homogenization. Therefore, there is a very significance in studying the large-time behavior of solutions for Hamilton-Jacobi equations.This paper is concerned with the large-time behavior of solutions for Hamilton-Jacobi equations.Chapter 1 introduces the general knowledge, includes the main works,several important definitions and theorems and so on.In chapter 2, we study the long-time behavior of viscosity solutions of Hamilton-Jacobi equations by using the generalized dynamical approach, and establish the general convergence result of the Cauchy problem for Hamilton-Jacobi equations.In chapter 3,by using the PDE method, we study the large-time behavior of viscosity solutions of Hamilton-Jacobi equations. And we establish the general convergence result of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations.In chapter 4, we generalize the comparison theorems of Hitoshi Ishii[10] and Hiroyoshi Mitake[20] on autonomous of Hamilton-Jacobi equations to the case of time-periodic Hamilton-Jacobi equations.