Exponential Scattering for a Damped Hartree Equation

This note studies the linearly damped generalized Hartree equation iu˙−(−Δ)su+iau=±|u|p−2(Jγ∗|u|p)u,0<s<1,a>0,p≥2. Indeed, one proves an exponential scattering of energy global solutions, with spherically symmetric datum. means that, for large time, solution goes exponentially to associated free problem iu˙−(−Δ)su+iau=0, in Hs norm. The radial assumption avoids a loss regularity Strichartz estimates. scattering, which that v:=eatu scatters Hs, is proved sub-critical defocusing regime and mass-sub-critical focusing regime. result presented because gap due lack mass regime, seems not be well understood. In this manuscript, needs overcome three technical difficulties are mixed together: first fractional Laplace operator, second Choquard (non-local) source term, including Hartree-type term when p=2 last damping iau. work progress, authors investigate solutions above Schrödinger problem, different kind terms.