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

Abstract

We investigate the well-posedness in the generalized Hartree equation iut + ∆u + (|x| −(N−γ) * |u| p )|u| p−2 u = 0, x ∈ R N , 0 < γ < N , for low powers of nonlinearity, p < 2. We establish the local wellposedness 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.