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Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS


Authors: Pierre Raphaël and Jeremie Szeftel
Journal: J. Amer. Math. Soc. 24 (2011), 471-546
MSC (2010): Primary 35B35, 35B44; Secondary 35Q41, 35Q55
DOI: https://doi.org/10.1090/S0894-0347-2010-00688-1
Published electronically: December 2, 2010
MathSciNet review: 2748399
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Abstract: We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $ i\partial_tu+\Delta u+k(x)\vert u\vert^{2}u=0$. From a standard argument, there exists a threshold $ M_k>0$ such that $ H^1$ solutions with $ \Vert u\Vert _{L^2}<M_k$ are global in time while a finite time blow-up singularity formation may occur for $ \Vert u\Vert _{L^2}>M_k$. In this paper, we consider the dynamics at threshold $ \Vert u_0\Vert _{L^2}=M_k$ and give a necessary and sufficient condition on $ k$ to ensure the existence of critical mass finite time blow-up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow-up elements at a nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case $ k\equiv 1$.


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Additional Information

Pierre Raphaël
Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Email: pierre.raphael@math.univ-toulouse.fr

Jeremie Szeftel
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75 005 Paris, France
Email: jeremie.szeftel@ens.fr

DOI: https://doi.org/10.1090/S0894-0347-2010-00688-1
Received by editor(s): January 4, 2010
Received by editor(s) in revised form: September 2, 2010
Published electronically: December 2, 2010
Additional Notes: The first author was supported by the French Agence Nationale de la Recherche, ANR jeunes chercheurs SWAP and by ANR OndeNonLin
The second author was supported by the French Agence Nationale de la Recherche, ANR jeunes chercheurs SWAP
Article copyright: © Copyright 2010 American Mathematical Society

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