Study On Asymptotic Behavior Of Viscosity Solutions For Hamilton-jacobi Equations

Hamilton-Jacobi equations arise in classical mechanics and geometric optics. As first order nonlinear partial differential equations, Hamilton-Jacobi equation is one of the most important mathematical models of ocean internal wave dynamics, fluid mechanics and atmospheric dynamics, and it has very important applications in Hamilton Dynamics, optimal control theory and differential games. The classical smooth solution of Hamilton-Jacobi equation is not easy to find or even not exist because of development of shocks in solutions. It is necessary to solve this problem for the development of many application disciplines. At the beginning of 1980s, Crandall and Lions introduced the notion of viscosity solution in the study ofHamilton-Jacobi equations based on the maximum principle, which had greatly advanced the development of weak solutions theory of partial differential equations. Homogenization theory of dynamical system was put forward by Gauss when he study the mutual perturbation of planets, which is classical analytic method in the study of the asymptoticity of solutions of partial differential equations or dynamical system with multiscale structure. After the viscosity solutions defined by Lions, Crandall and Evans, the research in Hamilton Dynamics has entered a brand-new field. Because of the essence of contact between viscosity solutions, weak KAM theory and homogenization theory of Hamilton system, it is important to study homogenization theory.The study of the large-time behavior of solutions of the Cauchy problem for Hamilton-Jacobi equations goes back to the works of Kruzkov, Lions, and Barles, who studied the case whenΩ=Rn anddoes not depend on the variable x . At present, a characteristic of the research and development for the asymptotic behavior of the solutions is based on weak KAM theory introduced by Fathi. And this theory play an important role on the research of the solution's structure of the station- ary equation. There are also some results on the large-time asymptotic problem which treat Hamilton-Jacobi equations with boundary conditions.The theory of almost periodic functions was first developed on the research of Fourier series by the Danish mathematician Bohr during 1924-1926. Because of practical problems and the promotion of other branches of mathematics, the theory of almost periodic differential equations have considerable development. In developing the theory of almost periodic functions, a main feature is that the scope is expanding. From almost periodic function, asymptotically almost periodic function, weakly almost periodic function, remotely almost periodic function introduced by Sarason in 1984 to pseudo almost periodic function proposed by Chuanyi Zhang in 1992. Every extension enlarges the application of the almost periodic type functional theory.This paper is composed of five chapters and the main results are described as follows:In Chapter One, we introduce the background, the development and overview of Hamilton-Jacobi equations, homogenization, large time behavior of solutions and almost periodic type functions.In Chapter Two, we elaborate elementary knowledge of Hamilton-Jacobi equations in detail.In Chapter Three, we use the notion of transverlity between curves and sets of vanishing Lebesgue measure, the use of a maximum principle and the perturbed test-function argument, and we study the behavior of viscosity solutions asε→0 of Cauchy problem for Hamilton-Jacobi equation the convergence of uε, to the solution of locally and uniformly in In Chapter Four, we investigate the large time behavior of viscosity solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations in a bounded domain with the asymptotically almost periodic boundary conditions by Aubry-Mather theory and the Perron's metnod. We establish a convergence result for viscosity solutions as time goes to infinity and the representation formulas of asymptotical solutions.In Chapter Five, we study on the remotely almost periodic solutions of first order differential equations involving reflection of the argument and the almost periodic solutions of second order equations by the basic definition and properties of almost periodic type functions and Banach contraction mapping priniple, showing the existence and uniqueness of the solution.