  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 ($3\leq p<5$) whose initial data are radial and come with a finite
energy. In this work we prove scattering in the positive time direction by only
assuming the inward part of the energy decays at a certain rate, as long as the
total energy is finite, regardless of the decay rate or size of the outward
energy. More precisely, we assume the initial data comes with a finite energy
and \[
  \int_{{\mathbb R}^3} \max\{1,|x|^\kappa\}\ (\ |\nabla u_0(x)\cdot
\frac{x}{|x|} + \frac{u_0(x)}{|x|} + u_1(x)\ |^2 + \frac{2}{p+1}|u_0(x)|^{p+1}\
) dx < \infty. \] Here $\kappa\geq \kappa_0(p) = \frac{5-p}{p+1}$ is a
constant. If $\kappa>\kappa_0(p)$, we can also prove $\|u\|_{L^p
L^{2p}}({\mathbb R}^+ \times {\mathbb R}^3)< +\infty$ and give an explicit rate
of $u$'s convergence to a free wave.
