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

@article{Raphael2010ExistenceAU, title={Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS}, author={Pierre Raphael and J{\'e}r{\'e}mie Szeftel}, journal={Journal of the American Mathematical Society}, year={2010}, volume={24}, pages={471-546} }

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…

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