In this paper we consider 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 $p \in [3,5)$. 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 $\varepsilon > 0$, then the corresponding solution $u$ must exist for all time $t \in { \mathbb R}$ 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.
Categories: math.AP