On the existence time for the Kirchhoff equation with periodic boundary conditions

Abstract

We consider the Cauchy problem for the Kirchhoff equation on ${\mathbb{T}}^d$ with initial data of small amplitude $\epsilon$ in Sobolev class. We prove a lower bound ${\epsilon }^{-4}$ for the existence time, which improves the bound ${\epsilon }^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.

30 pages