An inhomogeneous, $L^2$ critical, nonlinear Schrödinger equation

An inhomogeneous nonlinear Schrödinger equation is considered, that is invariant under $L^2$ scaling. The sharp condition for global existence of $H^1$ solutions is established, involving the $L^2$ norm of the ground state of the stationary equation. Strong instability of standing waves is proved by constructing self-similar solutions blowing up in finite time.

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