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The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov, where the pathological set contains a superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schrödinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity, which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales.

Pathological Set of Initial Data for Scaling-Supercritical Nonlinear Schrödinger Equations