Homogenization Of Nonautonomous Hamilton-Jacobi Equations In Stationary Ergodic Setting

In this research we vague the difference between the time and spatial variableto generate the corresponding results of Lions and Souganidis [37] to the nonautonomous case. We investigate the asymptotic behavior of the viscosity solutions of the Cauchy problem of Hamilton-Jacobi equationsWe gave the variational construction of the effective Hamiltonian and the sufficientcondition of the existence of the first order corrector. The effective Hamiltoniancomes down to some asymptotic behavior(lim_{ε→0}-εu-ε(u_t)²=(?)) of the viscosity solution of equationεu+ε(u_t)² + H(Du, x, t) = 0. When investigate the existence of the first order corrector we introduce another subjunctive time variable s and the corrector of the homogenization problem of equation( 1) comes down to some long time behavior (v(x, t) = lim_{s→∞} u(x, t, s,ω)) of the evolutionaryequation u_x(x,t,s) + H(Du,x,t,ω) = 0. Moreover using the corrector we also proved the viscosity solution of equation(1) will convergent to the viscosity solution of equationThe advantages of using the corrector method to prove the convergence theorem is that when a bounded corrector exists the proof becomes very easy.This paper is organized as follows, In chapter 2, we review some basic theoryof viscosity solutions and homogenization of Hamilton-Jacobi equation in stationary ergodic media. In chapter 3 we give the main results concerning the homogenization of equation(1) and in chapter 4 we give some applications.