Scattering for Defocusing NLS with Inhomogeneous Nonlinear Damping and Nonlinear Trapping Potential

We investigate an energy-subcritical defocusing nonlinear Schrödinger equation in $\mathbb R^3$ subject to a lower order nonlinear trapping potential and a spatially dependent nonlinear damping: \begin{equation*} i\partial_t u + Δu + i a(x) |u|^{2σ_2} u = |u|^{2σ_1} u + V(x)|u|^{2σ_3} u. \end{equation*} We prove that if the damping acts where $V$ induces concentration effects, i.e. where $V$ is either negative or non-repulsive, solutions are global and uniformly bounded in $H^1$, and scatter in the intercritical regime. A primary challenge arises from the spatial dependence of $a(x)$, which breaks the energy's monotonicity. Consequently, a uniform in time control of the $H^1$ norm of a solution is non-trivial and represents a new result even for $V = 0$. We overcome this issue by introducing a novel energy modified by virial argument, showing simultaneously a uniform bound on the energy and local energy decay estimates, which are subsequently upgraded to scattering via interaction Morawetz estimates.