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# Pure and Applied Mathematics Quarterly

## Volume 16 (2020)

### Number 5

### A Nekhoroshev type theorem for the nonlinear wave equation on the torus

Pages: 1739 – 1765

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a14

#### Authors

#### Abstract

In this paper, we prove a Nekhoroshev type theorem for the nonlinear wave equation\[u_{tt} = u_{xx} - mu - f (u)\]on the finite $x$-interval $[0, \pi]$. The parameter m is real and positive, and the nonlinearity $f$ is assumed to be real analytic in $u$. More precisely, we prove that if the initial datum is analytic in a district of width $2 \rho \gt 0$ whose norm on this district is equal to $\varepsilon$, then if $\varepsilon$ is small enough, the solution of the nonlinear wave equation above remains analytic in a district of width $\rho / 2$, with norm bounded on this district by $C \varepsilon$ over a very long time interval of order $\varepsilon^{- \sigma {\lvert \: \mathrm{lm} \: \varepsilon \: \rvert}^\beta}$, where $0 \lt \beta \lt 1/7$ is arbitrary and $C \gt 0$ and $\sigma \gt 0$ are positive constants depending on $\beta$ and $\rho$.

#### Keywords

wave equation, Birkhoff normal form, long time stability

#### 2010 Mathematics Subject Classification

Primary 37J40, 37K55. Secondary 35B35, 35Q35.

The first-named author is supported by Shandong Provincial Natural Science Foundation No. ZR2019MA062 and Binzhou University (BZXYL1402).

Received 28 August 2017

Accepted 21 December 2018

Published 17 February 2021