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arXiv · 2309.08281

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

Paolo Antonelli·Boris Shakarov
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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.

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Paolo Antonelli, Boris Shakarov. 2023-09-15. On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping. https://arxiv.org/abs/2309.08281

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