# Pure and Applied Mathematics Quarterly

## Volume 16 (2020)

### 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

Lufang Mi (College of Science, Binzhou University, Shandong, China)

Chunyong Liu (Shandong Chemical Engineering and Vocational College, Weifang, Shandong, China)

Guanghua Shi (College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China)

Rong Zhao (Dalian Airforce Communication Non-Commissioned Officer Academy, Liaoning, China)

#### 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).