Abstract
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities
When \(p_1=\frac{4}{N}\) and \(0<p_2<\frac{4}{N}\), we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for this equation. This is a complement to the result of Tao et al. (Commun Partial Differ Equ 32:1281–1343, 2007). Moreover, we investigate the dynamical properties of blow-up solutions, including \(L^2\)-concentration, blow-up rates and limiting profile. When \(\frac{4}{N}<p_1<\frac{4}{N-2}\)(\(4<p_1<\infty \) if \(N=1\), \(2<p_1<\infty \) if \(N=2\)), we prove that the blow-up 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.
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This work is supported by NSFC Grants (Nos. 11601435, 11401478), Gansu Provincial Natural Science Foundation (1606RJZA010) and NWNU-LKQN-14-6.
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Feng, B. On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities. J. Evol. Equ. 18, 203–220 (2018). https://doi.org/10.1007/s00028-017-0397-z
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DOI: https://doi.org/10.1007/s00028-017-0397-z