# Mathematics Colloquia and Seminars

Abstract: A basic question in the theory of nonlinear hyperbolic equations is the following: given initial data of size $\varepsilon$, where $\varepsilon << 1$ is a small parameter, what is the lifespan $t = t(\varepsilon)$ of the corresponding solution? We investigate this question in the context of the nonlinear Klein-Gordon equation with polynomial nonlinearity. Standard energy estimates yield that solutions to the quadratic and cubic equations have a life-span $t = \mathcal{O}(1/\varepsilon)$ and $t = \mathcal{O}(1/\varepsilon^2)$, respectively. The method of normal forms, introduced by Shatah (1985) and based upon Poincare's method of normal forms for ODEs, allows one to transform the quadratic nonlinearity into a cubic nonlinearity, thereby extending the lifespan from $\mathcal{O}(1/\varepsilon)$ to $t = \mathcal{O}(1/\varepsilon^2)$. In this talk, we provide an introduction to the method of normal forms in the context of the nonlinear Klein-Gordon equation with quadratic nonlinearity. This talk is the first in a series of talks based around the method of normal forms.