A note on concentration for blowup solutions to supercritical Schrödinger equations

Guo, Qing

doi:10.1090/S0002-9939-2013-11735-1

A note on concentration for blowup solutions to supercritical Schrödinger equations
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Proc. Amer. Math. Soc. 141 (2013), 4215-4227 Request permission

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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