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

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

2010-01-11

arxiv.org/abs/1001.1627v1

Mathematics

Analysis of PDEs

Scientific paper

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 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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