Abstract
This note studies the focusing generalized Hartree equation
i(u) over dot + Delta u + vertical bar u vertical bar(p-2) (J(rho) * vertical bar u vertical bar(p))u = 0,
in the inter-critical-regime with spherically symmetric data. Using a scattering/blow-up criteria, one revisits the scattering versus blow-up of energy solutions. One considers mainly the dynamics of energy solutions in three cases: below, at and beyond the ground state threshold. The scattering proof uses a new method due to Dodson and Murphy (Proc Am Math Soc 145(11), 4859-4867, 2017). Here, one gives some simplified proofs of the scattering threshold already obtained by the second author Saanouni (J Math Anal Appl 492(1), 124436, 2020) and (NoDEA 26, 41, 2019). The blow-up criteria is based on a localized variance identity and gives a non global existence or an infinite time blow-up parallel to del u(t(n))parallel to(L2(RN)) -> infinity, with t(n) -> infinity. In some results, one needs to assume the uniqueness of spherically symmetric positive ground states, which seems to be an open problem except for some particular cases.