The refined diagonalization procedure and a development on the evolution equations with smooth coefficients

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In this project, we studied evolution equations with smooth variable coefficients by using a special technique for the precise representations of the solutions, which is called "the refined diagonalization procedure". This method is very useful for the analysis of the initial values problems of hyperbolic equations with variable coefficients, for example, wave equations, Klein-Gordon type equations, second order hyperbolic equations and semi-linear wave equations. Indeed, we succeeded to prove some estimates for these problems, which couldn't be solved by using the previous method, by using our new method. Moreover, we expect that our results will be applicable for the analysis of non-linear problems, in particular, for the global solvability of Kirchhoff equation in non-realanalytic class.