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

从高频渐近分析的角度研究了双曲型和色散型非线性偏微分方程解的稳定性。揭示了非线性Klein-Gordon方程组的零结构与共振相互作用之间的关系。构造了非线性薛定谔方程小数据爆破的例子，并给出了爆破解的寿命阶的估计。并将高频渐近方法应用于一类不满足零条件的非线性波动方程。

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.