Merle, Frank; Raphaël, Pierre; Rodnianski, Igor Type II blow up for the energy supercritical NLS. (English) Zbl 1347.35215 Camb. J. Math. 3, No. 4, 439-617 (2015). Let \(d\geq 11\) and \(p>p(d)\) be large enough. The purpose of this paper is to study the NLS \[ i \partial_t u+\triangle u+u| u|^{p-1}=0, \] an energy critical and super critical model. Instead the authors study the corresponding heat equation in dimension \(d\geq 3\), make a substitution, and obtain an elliptic stationary self-similar equation, with two well-known spherically symmetric solutions, with corresponding blow ups. The main result is that the asymptotics for \(p\geq pJL\), which replace a well-known expansion in the critical case, are perfectly suitable for the construction of a certain blow-up solution. The proofs use Lyapunov functional, maximum principle tools, Liouville classification theorem, Sturm-Liouville oscillation argument, Morawetz type stimates, Hardy type coercivity, high order Sobolev norms, and Green functions. Several long computations are collected in appendices at the end of the paper. Reviewer: Thomas Ernst (Uppsala) Cited in 2 ReviewsCited in 28 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B44 Blow-up in context of PDEs Keywords:NLS; critical and super critical model; heat equation; spherically symmetric solution PDFBibTeX XMLCite \textit{F. Merle} et al., Camb. J. Math. 3, No. 4, 439--617 (2015; Zbl 1347.35215) Full Text: DOI arXiv