Scattering theory for the radial H˙12-critical wave equation with a cubic convolution

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution ∂t2u−Δu=±(|x|−3⁎|u|2)u in dimensions d≥4d≥4. We prove that if the radial solution u with life-span I   obeys (u,ut)∈Lt∞(I;H˙x1/2(Rd)×H˙x−1/2(Rd)), then u is global and scatters. By the strategy derived from concentration compactness, we show that the proof of the global well-posedness and scattering is reduced to disprove the existence of two scenarios: soliton-like solution and high to low frequency cascade. Making use of the No-waste Duhamel formula and double Duhamel trick, we deduce that these two scenarios enjoy the additional regularity by the bootstrap argument of [7]. This together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction.