Asymptotic behavior and life-span estimates for the damped inhomogeneous nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the lifespan and asymptotic behavior of solutions to a damped inhomogeneous nonlinear Schrödinger equation, providing bounds, blow-up conditions, and scattering results, including new findings for specific parameter cases.

Contribution

It introduces explicit lifespan bounds, blow-up criteria, and scattering results for the damped inhomogeneous nonlinear Schrödinger equation, extending known results to new parameter regimes.

Findings

Lifespan bounds for solutions

Conditions for blow-up and global existence

Decay rates and scattering behavior

Abstract

We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schr\"odinger equation $ i\partial_tu+\Delta u+\mu|x|^{-b}|u|^{\alpha}u+iau=0$, $\mu \in\mathbb{C} $, $b>0$, $a \in \mathbb{C}$ such that $\Re \textit{e}(a) \geq 0$, $\alpha>0$. We establish lower and upper bound estimates of the life-span. In particular for $a\geq 0$, we obtain explicit values $a_*,\; a^*$ such that if $a<a_*$ then blow up occurs, while for $a>a^*,$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$