Scattering for the Damped Focusing Nonlinear Schrödinger Equation in the Mass–Energy Intercritical Regime

We study the focusing nonlinear Schrödinger equation with time-dependent linear damping in the mass-supercritical and energy-subcritical regime. After a gauge renormalization, the problem becomes a focusing Schrödinger equation with a time-dependent coefficient in the nonlinearity. Under a natural dissipativity assumption on this coefficient, we prove forward scattering for radial H1 data below the ground-state threshold in dimensions d≥3. In contrast to previous scattering results based on a strictly positive averaged damping rate or on specially oscillatory initial data, we allow the averaged damping rate to vanish and require neither an oscillatory-data assumption nor a perturbative smallness condition. The proof combines variational trapping for the renormalized flow, localized coercivity, a localized Morawetz estimate, energy evacuation, and a radial scattering criterion adapted to the time-dependent setting.