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

TL;DR

This paper investigates how damping, inverse-square potential, and inhomogeneity affect the global existence and scattering of solutions to a nonlinear Schrödinger equation, providing new insights into long-time dynamics in this complex setting.

Contribution

It introduces the first comprehensive analysis of the combined effects of damping, inverse-square potential, and inhomogeneity on the NLS equation's solutions.

Findings

Global well-posedness established in energy space for various regimes.

Large damping influences solution dynamics significantly.

Extends understanding of long-time behavior in complex NLS models.

Abstract

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.