论文标题
Kirchhoff方程许多解决方案的寿命更长
Longer lifespan for many solutions of the Kirchhoff equation
论文作者
论文摘要
我们考虑kirchhoff方程$ \ partial_ {tt} u-ΔU\ big（1 + \ int _ {\ Mathbb t^d} | \ nabla u |^2 \ nabla u |^2 \ big）= 0 $ $ $ d $ d $ - d $ - d $ - d $ d $ \ mathbb t^d $及其cauck $ and cauch $ and cairatiate $ tirst in copliate y Inlart y（0） u（0，x）$ $ \ varepsilon $在sobolev类中。在以前的论文中以quasilarear formor形式获得的动力学的有效方程式在能量估计中包含与非平凡项相对应的共振。无法通过调整外部参数来避免这种共振（仅仅是因为Kirchhoff方程不包含参数）。 在本文中，我们在库奇问题的初始数据上介绍了非共振条件，并证明了相应解决方案的寿命下的下限$ \ varepsilon^{ - 6} $（标准的本地理论给出了$ \ varepsilon^{ - 2} $，并且分数的正常形式给出了$ \ varepsilon^$ \ varepsil^$ 4}。证据依赖于这样一个事实，即在这些非共振条件下，有效方程在很大的时间间隔中的“超级行动”的增长率比基于立方术语的正常形式的先验估计值较小（通过因子$ \ varepsilon^2 $）。满足这种非共振条件的初始数据集包含了本文中讨论的几个非平凡示例。
We consider the Kirchhoff equation $$ \partial_{tt} u - Δu \Big( 1 + \int_{\mathbb T^d} |\nabla u|^2 \Big) = 0 $$ 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 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.
