Scattering theory for the radial (H) over dot1/2-critical wave equation with a cubic convolution

[摘要] In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution partial derivative(2)(t)u - Delta u = +/-(vertical bar x vertical bar(-3) * vertical bar u vertical bar(2))u in dimensions d >= 4. We prove that if the radial solution u with life-span I obeys (u,u(t)) is an element of L-t(infinity))(I; (H) over dot(1/2)(R-d) x (H) over dot(1/2) (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 [7]. This together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction. (C) 2015 Elsevier Inc. All rights reserved.

[发布日期] 2015-12-15

[关键词] Wave-Hartree equation;Scattering theory;Concentration compactness