For slightly mass-supercritical NLS in 1 to 10 dimensions, no unstable high-energy eigenmodes exist for the linearized self-similar profile operator, which settles its asymptotic stability.
Mode stability for self-similar blowup of slightly supercritical nls: I
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Mode stability for self-similar blowup of slightly supercritical NLS: II. high-energy spectrum
For slightly mass-supercritical NLS in 1 to 10 dimensions, no unstable high-energy eigenmodes exist for the linearized self-similar profile operator, which settles its asymptotic stability.