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 $\rho$.

中文翻译：

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环上非线性波动方程的Nekhoroshev型定理

在本文中，我们证明了在有限$ x $-间隔$ [0，\ pi]上的非线性波动方程\ [u_ {tt} = u_ {xx}-mu-f（u）\]的Nekhoroshev型定理。 $。参数m是实数和正数，并且在$ u $中假定非线性度$ f $是实数解析。更确切地说，我们证明，如果初始数据是在宽度为$ 2 \ rho \ gt 0 $的区域中分析的，该区域在该区域上的范数等于$ \ varepsilon $，则如果$ \ varepsilon $足够小，则解上面的非线性波动方程仍然在$ \ rho / 2 $宽度的区域中进行分析，在该区域上的范数在$ \ varepsilon ^ {-\ sigma {\ lvert \：\ mathrm {lm} \：\ varepsilon \：\ rvert} ^ \ beta} $，其中$ 0 \ lt \ beta \ lt 1/7 $是任意的，而$ C \ gt 0 $和$ \ sigma \ gt 0 $是正常数，取决于$ \ beta $和$ \ rho $。