Damping Effects on Global Existence and Scattering for an Inhomogeneous NLS Equation with Inverse-Square Potential

This work explores the global existence and scattering behavior of solutions to a damped, inhomogeneous nonlinear Schrodinger equation featuring a time-dependent damping term, an inverse-square potential, and an inhomogeneous nonlinearity. We establish global well-posedness in the energy space for subcritical, mass-critical, and energy-critical regimes, using Strichartz estimates, Hardy inequalities, and Gagliardo-Nirenberg-type estimates. For sufficiently large damping, we highlight how the interplay between damping, singular potentials, and inhomogeneity influences the dynamics. Our results extend existing studies and offer new insights into the long-time behavior of solutions in this more general setting. To the best of our knowledge, this is the first study to address the combined effects of inverse-square potential, inhomogeneous (or even homogeneous) nonlinearity, and damping in the context of the NLS equation.