F. Genoud
Sharp global existence condition and instability by blowup for an inhomogeneous $L^2$ critical nonlinear Schrodinger equation
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ABSTRACT.  An inhomogeneous nonlinear Schrodinger equation is considered, which 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.