We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

中文翻译：

---

NLW解在-维环面上的长期行为

我们考虑非线性波动方程（NLW） $d$ 维环面 $\mathbb{T}^{d}$ 在原点处具有至少 2 阶的平滑非线性。我们证明，对于几乎任何质量，高 Sobolev 指数的小而平滑的解相对于初始数据的大小在任意长时间内都是稳定的。为了证明这一结果，我们使用范式变换将动力学分解为具有弱相互作用的低频和高频。虽然动力学的低部分可以置于经典的 Birkhoff 范式下，但高模式根据时间相关的线性哈密顿系统演化。然后，我们通过对高模式使用多项式增长估计和对低模式保留 Sobolev 范数来控制全局动态。我们的一般策略适用于任何线性频率满足非常一般的非共振条件的半线性哈密顿偏微分方程 (PDE)。 $\mathbb{T}^{d}$ 是一个很好的例子，因为标准 Birkhoff 范式仅适用于 $d=1$ 而我们的策略适用于任何方面。