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

2018/05/03 by Pietro Baldi, Baldi, Pietro, Emanuele Haus +1 · 1 citation
Engineering · Mathematics · #35L15 #35L72 #35Q74 #37J40 #70K45 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1805.01189

openalex publication_date 2018/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem for the Kirchhoff equation on mathbbT^d\nwith initial data of small amplitude \ε in Sobolev class. We prove a\nlower bound \ε^-4 for the existence time, which improves the bound\n\ε^-2 given by the standard local theory. The proof relies on a\nnormal form transformation, preceded by a nonlinear transformation that\ndiagonalizes the operator at the highest order, which is needed because of the\nquasilinear nature of the equation.\n

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