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arXiv · 1805.05688

Blow-up for the nonlinear Schr\"{o}dinger equation with combined nonlinearities

Qing Guo·Shihui Zhu
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Abstract

In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schr\"{o}dinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type. Especially, we find a sequence of initial data with mass approximating that of the ground state from above, the correspondng solution of which blows up. This result partially gives a positive answer to the open problem left by T.Tao, M. Visan and X. Zhang in \cite{TVZ}. After then, we obtain the lower blow-up rate and concentration rate for the blow-up solutions in the same case. Finally, we consider the mass-critical combined with the mass-supercritical power type case, studying the blow-up criteria, blow-up rate and concentration of the blow-up solutions in this case.

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Qing Guo, Shihui Zhu. 2018-05-15. Blow-up for the nonlinear Schr\"{o}dinger equation with combined nonlinearities. https://arxiv.org/abs/1805.05688

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