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Abstract

We study the blow-up of solutions to the focusing Hartree equation $iu_t+ \Delta u+(|x|^{-\gamma }*|u|^2)u=0$. We use the strategy derived from the almost finite speed of propagation ideas devised by Bourgain (1999) and virial analysis to deduce that the solution with negative energy ($E(u_0)<0$) blows up in either finite or infinite time. We also show a result similar to one of Holmer and Roudenko (2010) for the Schrödinger equations using techniques from scattering theory.