Asymptotic analysis on solutions to weakly coupled systems for fully nonlinear equations

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The main purpose is to deeply understand the structure of solutions to the weakly coupled system of fully nonlinear equations. More precisely, I mainly focused on some asymptotic problems (homogenization and large-time behavior), and got new results as follows. 1. Homogenization of the weakly coupled system of Hamilton-Jacobi equation with fast switching rates, analysis on the initial layer and the effective Hamiltonian. 2. Large-time behavior of the same system by analyzing the value functions of the optimal control problems which appear in the dynamic programming for the system whose states are governed by random changes. 3. Large-time asymptotics for degenerate viscous Hamilton-Jacobi equations by using the nonlinear adjoint method, which was completely new and giving a breakthrough on the study of the large-time asymptotics for degenerate parabolic equations.