Minimal blow-up initial data for potential blow-up solutions to inter-critical Schrödinger equations

ABSTRACT

We show that assuming some initial datum $u_0\in \dot{\phantom{a}}{H}^{s_c}\left({\mathbb{R}}^d\right)$ leads to a blow-up solution of the inter-critical defocusing nonlinear Schrödinger equation, there must exist an initial datum with the minimal ${\dot{H}}^{s_c}$-norm which will produce blow-up. Moreover, up to the invariant transformations, the set of such data is compact in ${\dot{H}}^{s_c}\left({\mathbb{R}}^d\right)$. The main tool is the profile decomposition combined with the perturbation argument.

KEYWORDS:

• Nonlinear Schrödinger equation
• inter-critical
• blow-up
• profile decomposition

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 35Q55

Acknowledgments

The authors would like to express their gratitude to the referees for useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (nos. 11301564, 11771469).