The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $3\leq p \frac{5-p}{2}$, then the solution scatters as long as the initial data $(u_0,u_1)$ satisfy \[ \int_{{\mathbb R}^3} (|x|^\kappa+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u_0|^{p+1}\right) dx < +\infty. \] If $p=3$, we can also prove the scattering result if initial data $(u_0,u_1)$ are contained in the critical Sobolev space and satisfy the inequality \[ \int_{{\mathbb R}^3} |x|\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{4}|u_0|^{p+1}\right) dx < +\infty. \] These assumptions on the decay rate of initial data as $|x| \rightarrow \infty$ are weaker than previously known scattering results.

中文翻译：

---

3D半线性波动方程非径向解的内向/外向能量理论

这篇论文的题目是一个半线性、能量次临界、离焦波动方程$\partial_t^2 u - \Delta u = - |u|^{p -1} u$在3维空间中，$3 \leq p \frac{5-p}{2}$，那么只要初始数据 $(u_0,u_1)$ 满足 \[ \int_{{\mathbb R}^3} (|x| ^\kappa+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}| u_0|^{p+1}\right) dx < +\infty。\] 如果$p=3$，我们也可以证明如果初始数据$(u_0,u_1)$ 包含在临界Sobolev 空间中并且满足不等式\[ \int_{{\mathbb R}^3} |x|\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{4}|u_0|^{p +1}\right) dx < +\infty。\] 这些对初始数据衰减率的假设为$|x| \rightarrow \infty$ 比之前已知的散射结果要弱。