In this paper we consider a semi-linear, energy sub-critical, defocusing waveequation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensionalspace with $p \in [3,5)$. We prove that if initial data $(u_0, u_1)$ are radialso that $\|\nabla u_0\|_{L^2 ({\mathbb R}^3; d\mu)}, \|u_1\|_{L^2 ({\mathbbR}^3; d\mu)} \leq \infty$, where $d \mu = (|x|+1)^{1+2\varepsilon}$ with$\varepsilon > 0$, then the corresponding solution $u$ must exist for all time$t \in {\mathbb R}$ and scatter. The key ingredients of the proof include atransformation $\mathbf{T}$ so that $v = \mathbf{T} u$ solves the equation$v_{\tau \tau} - \Delta_y v = - \left(\frac{|y|}{\sinh |y|}\right)^{p-1}e^{-(p-3)\tau} |v|^{p-1}v$ with a finite energy, and a couple of globalspace-time integral estimates regarding a solution $v$ as above.
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机译：在本文中，我们考虑了具有$ p \ in的3维空间中的半线性，能量次临界，散焦波动方程$ \ partial_t ^ 2 u-\ Delta u =-| u | ^ {p -1} u $ [3,5）$。我们证明，如果初始数据$（u_0，u_1）$是径向的，那么$ \ | \ nabla u_0 \ | _ {L ^ 2（{\ mathbb R} ^ 3; d \ mu）}，\ | u_1 \ | _ {L ^ 2（{\ mathbbR} ^ 3; d \ mu）} \ leq \ infty $，其中$ d \ mu =（| x | +1）^ {1 + 2 \ varepsilon} $ with $ \ varepsilon> 0 $，则相应的解决方案$ u $必须在{\ mathbb R} $中一直存在并且分散。证明的关键要素包括变换$ \ mathbf {T} $，以便$ v = \ mathbf {T} u $解方程$ v _ {\ tau \ tau}-\ Delta_y v =-\ left（\ frac { | y |} {\ sinh | y |} \ right）^ {p-1} e ^ {-（p-3）\ tau} | v | ^ {p-1} v $具有有限的能量，并且关于上述解决方案$ v $的几个全球时空积分估计。
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