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arXiv · 1805.01189

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

Pietro Baldi·Emanuele Haus
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Abstract

We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ 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.

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Pietro Baldi, Emanuele Haus. 2018-05-03. On the existence time for the Kirchhoff equation with periodic boundary conditions. https://arxiv.org/abs/1805.01189

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