High frequency asymptotic analysis for nonlinear PDEs of hyperbolic and dispersive type

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Nonlinear partial differential equations of hyperbolic and dispersive type are studied from the view point of high frequency asymptotic analysis. Relations between the null structure and resonant interactions are revealed for systems of nonlinear Klein-Gordon equations. Examples on small data blow-up for nonlinear Schrodinger equations are constructed, and estimates for the order of the lifespan of the blowing-up solutions are provided. The method of high frequency asymptotics are also applied to a class of nonlinear wave equations not satisfying the null condition.