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A note on concentration for blowup solutions to supercritical Schrödinger equations


Author: Qing Guo
Journal: Proc. Amer. Math. Soc. 141 (2013), 4215-4227
MSC (2010): Primary 35Q55, 35A15; Secondary 35B30
Published electronically: August 30, 2013
MathSciNet review: 3105865
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Abstract: We study the blowup dynamics of solutions to the $ L^2$-supercritical nonlinear Schrödinger equation and prove that the blowup solution with
bounded $ \dot H^{s_c}$ norm must concentrate at least a fixed amount of the $ \dot H^{s_c}$ norm and, also, its $ L^{p_c}$ norm must concentrate at least a fixed $ L^{p_c}$ norm. We show these properties without any further symmetry assumptions on the solution and partly generalize the results obtained in papers of Holmer and Roudenko and of Zhu, which only deal with the radially symmetric case. Our proof is based on the profile decomposition theorems.


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Additional Information

Qing Guo
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Address at time of publication: College of Science, Minzu University of China, Beijing 100081, People’s Republic of China
Email: guoqing@amss.ac.cn

DOI: https://doi.org/10.1090/S0002-9939-2013-11735-1
Keywords: Schr\"odinger equation, blowup solution, concentration, supercritical
Received by editor(s): May 3, 2011
Received by editor(s) in revised form: November 8, 2011, and January 8, 2012
Published electronically: August 30, 2013
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.