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

Raphaël, Pierre; Szeftel, Jeremie

doi:10.1090/S0894-0347-2010-00688-1

Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS
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by Pierre Raphaël and Jeremie Szeftel;

J. Amer. Math. Soc. 24 (2011), 471-546

DOI: https://doi.org/10.1090/S0894-0347-2010-00688-1

Published electronically: December 2, 2010

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Abstract:

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial _tu+\Delta u+k(x)|u|^{2}u=0$. From a standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow-up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ 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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