On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation |
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Authors: | Frank Merle Pierre Raphael |
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Affiliation: | Université de Cergy--Pontoise, Cergy-Pointoise, France, Institute for Advanced Studies and Centre National de la Recherche Scientifique Pierre Raphael ; Université de Cergy--Pontoise, Cergy-Pointoise, France, Institute for Advanced Studies and Centre National de la Recherche Scientifique |
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Abstract: | We consider the critical nonlinear Schrödinger equation with initial condition in the energy space and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in a sharp and stable upper bound on the blow-up rate: . In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, In this paper, we prove the sharp lower bound
by exhibiting the dispersive structure in the scaling invariant space for this log-log regime. In addition, we will extend to the pure energy space a dynamical characterization of the solitons among the zero energy solutions. |
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Keywords: | Critical Schr" odinger equation, finite time blowup, blow-up rate, log-log law. |
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