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Scattering theory for the radial $\dot H^{1/2}$-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 $\partial_{t}^2u-Δu=\pm(|x|^{-3}\ast|u|^2)u$ in dimensions $d\geq4$. We prove that if the radial solution $u$ with life-span $I$ obeys $(u,u_t)\in L_t^\infty(I;\dot H^{1/2}_x(\mathbb R^d)\times\dot H^{-1/2}_x(\mathbb R^d))$, 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 [Dodson,Lawrie, Anal.PDE, 8(2015), 467-497]. This together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction.