For radial finite-energy solutions of the 3D defocusing energy-subcritical nonlinear wave equation, decay of only the inward part of the energy implies forward scattering, and slightly faster decay gives explicit rates.
Scattering for radial, bounded solutions of focusing supercritical wave equations
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Scattering of solutions to NLW by Inward Energy Decay
For radial finite-energy solutions of the 3D defocusing energy-subcritical nonlinear wave equation, decay of only the inward part of the energy implies forward scattering, and slightly faster decay gives explicit rates.