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arXiv · 1001.1627

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

Pierre Raphael·Jeremie Szeftel
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Abstract

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+Δu+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2} M_k$. 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 non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$.

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Pierre Raphael, Jeremie Szeftel. 2010-01-11. Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS. https://arxiv.org/abs/1001.1627

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