Scattering of solutions to the defocusing energy sub-critical semi-linear wave equation in 3D

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u=-\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in the 3-dimensional space with $p\in [3,5)$. We prove that if initial data $(u_0,u_1)$ are radial so that $|\nabla u_0|{L^2 ({\mathbb R}^3; d\mu)}, |u_1|{L^2 ({\mathbb R}^3; d\mu)} \leq \infty$,where$d \mu = (|x|+1)^{1+2\varepsilon}$with$\varepsilon > 0$,thenthecorrespondingsolution$u$mustexistforalltime$t \in {\mathbb R}$ and scatter. The key ingredients of the proof include a transformation $\mathbf{T}$ 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 global space-time integral estimates regarding a solution $v$ as above.