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On uniqueness and continuation properties after blow‐up time of self‐similar solutions of nonlinear schrödinger equation with critical exponent and critical mass
Author(s) -
Merle Frank
Publication year - 1992
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160450204
Subject(s) - mathematics , uniqueness , continuation , nonlinear system , critical exponent , conformal map , mathematical analysis , norm (philosophy) , nonlinear schrödinger equation , schrödinger equation , exponent , scaling , physics , geometry , law , quantum mechanics , computer science , political science , programming language , linguistics , philosophy
We consider the nonlinear Schrödinger equation with critical power\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {(1)} \hfill & {} \hfill & {i\frac{{\partial u}}{{\partial t}} = \ \ - \Delta u - \left| u \right|^{4/N} u,} \hfill \\ {(1')} \hfill & {} \hfill & {u(0, \cdot) = \phi (\cdot),} \hfill \\ \end{array} $$\end{document} where u : (0, T ) × ℝ N → C and ø ϕ H 1 ∪ {ø;|x|ø ϵ L 2 }. With this nonlinear term, the equation (1)‐(1′) has a conformal invariance. Thus one yields explicit “self‐similar” solutions which have the following property: they have minimum L 2 norm among blow‐up solutions. In this paper, we first focus on uniqueness properties of these self‐similar solutions in a certain class. We then look at the possible continuations of these solutions after the blow‐up time.