For the one-dimensional nonlinear damped Klein–Gordon equation

$$\begin{aligned} \partial _{t}^{2}u+2\alpha \partial _{t}u-\partial _{x}^{2}u+u-|u|^{p-1}u=0 \quad \text{ on } \mathbb {R}\times \mathbb {R}, \end{aligned}$$

with \(\alpha >0\) and \(p>2\), we prove that any global finite energy solution either converges to 0 or behaves asymptotically as \(t\rightarrow \infty \) as the sum of \(K\ge 1\) decoupled solitary waves. In the multi-soliton case \(K\ge 2\), the solitary waves have alternate signs and their distances are of order \(\log t\).

中文翻译：

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一维阻尼非线性Klein-Gordon方程的长时间渐近性

对于一维非线性阻尼Klein-Gordon方程

$$ \ begin {aligned} \ partial _ {t} ^ {2} u + 2 \ alpha \ partial _ {t} u- \ partial _ {x} ^ {2} u + u- | u | ^ {p -1} u = 0 \ quad \ text {on} \ mathbb {R} \ times \ mathbb {R}，\ end {aligned} $$

通过\（\ alpha> 0 \）和\（p> 2 \），我们证明任何全局有限能量解都收敛为0或渐近地表现为\（t \ rightarrow \ infty \）作为\（K \ ge 1 \）解耦的孤立波。在多孤子情况\（K \ ge 2 \）中，孤立波具有交替的符号，其距离为\（\ log t \）。