Longer Lifespan for Many Solutions of the Kirchhoff Equation

We consider the Kirchhoff equation $\partial_{tt} u - \Delta u \big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \big) = 0$ on the $d$-dimensional torus $\mathbb{T}^d$, and its Cauchy problem with initial data $u(0,x)$, $\partial_t u(0,x)$ of size $\varepsilon$ in the Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound $\varepsilon^{-6}$ for the lifespan of the corresponding solutions (the standard local theory gives $\varepsilon^{-2}$, and the normal form for the cubic terms gives $\varepsilon^{-4}$). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the “superactions” of the effective equations on large time intervals is smaller (by a factor $\varepsilon^2$) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper.