Scattering of solutions to NLW by Inward Energy Decay

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.