Scattering theory for the radial ${\dot{H}}^{1/2}$-critical wave Equation with a cubic convolution

Abstract

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution ${\partial }_t^{2}u-\Delta u=\pm (|x{|}^{-3}\ast |u{|}^2)u$ in dimensions $d\ge 4$. We prove that if the radial solution $u$ with life-span $I$ obeys $(u,u_t)\in L_t^{\infty }(I;{\dot{H}}_x^{1/2}({\mathbb{R}}^d)\times {\dot{H}}_x^{-1/2}({\mathbb{R}}^d))$, 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.

32 pages

References in corpus (5)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Scattering for the radial 3d cubic wave equation
• Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$-subcritical Hartree equation in ${\mathbb{R}}^d$
• Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm
• The radial defocusing nonlinear Schrödinger equation in three space dimensions

Cited by in corpus (2)

• Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation
• Global Cauchy problems for the Klein-Gordon, wave and fractional Schrödinger equations with Hartree nonlinearity on modulation spaces