Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$

Abstract

We investigate the well-posedness in the generalized Hartree equation $i{u}_t+\Delta u+(|x{|}^{-(N-\gamma )}\ast |u{|}^p)|u{|}^{p-2}u=0$, $x\in {\mathbb{R}}^N$, $0<\gamma <N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin [6]. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

29 pages, accepted version