Existence and nonexistence of global solutions for time-dependent damped NLS equations

TL;DR

This paper studies the nonlinear Schrödinger equation with time-dependent damping, establishing conditions for solution blow-up or global existence, thus advancing understanding of damping effects on solution behavior.

Contribution

It generalizes previous results by analyzing non-standard damping assumptions, providing new criteria for blow-up and global solutions in different regimes.

Findings

Blow-up occurs in the inter-critical regime under certain damping conditions.

Global existence is proven in the energy subcritical case with specific assumptions.

Results extend and improve previous findings in the literature.

Abstract

We investigate the Cauchy problem for the nonlinear Schr\"odinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global existence in the energy subcritical case. Our results generalize and improve the ones in [9, 11, 21].