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Type II blow up for the energy supercritical NLS. (English) Zbl 1347.35215

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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
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