On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping

TL;DR

This paper rigorously demonstrates that nonlinear damping does not prevent finite-time singularity formation in the mass-supercritical focusing nonlinear Schrödinger equation, under specific conditions on damping strength and nonlinearity.

Contribution

It provides the first rigorous proof that nonlinear damping can still allow blow-up in the supercritical NLS, confirming previous numerical and formal predictions.

Findings

Singularities form for focusing and dissipative nonlinearities of the same power with small damping.

Damping can be controlled to preserve the self-similar blow-up structure.

Error terms are estimated via modulation analysis and control of energy and momentum.

Abstract

We consider the focusing, mass-supercritical NLS equation augmented with a nonlinear damping term. We provide sufficient conditions on the nonlinearity exponents and damping coefficients for finite-time blow-up. In particular, singularities are formed for focusing and dissipative nonlinearities of the same power, provided that the damping coefficient is sufficiently small. Our result thus rigorously proves the non-regularizing effect of nonlinear damping in the mass-supercritical case, which was suggested by previous numerical and formal results. We show that, under our assumption, the damping term may be controlled in such a way that the self-similar blow-up structure for the focusing NLS is approximately retained even within the dissipative evolution. The nonlinear damping contributes as a forcing term in the equation for the perturbation around the self-similar profile, that may produce a growth over finite time intervals. We estimate the error terms through a modulation analysis and a careful control of the time evolution of total momentum and energy functionals.