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

TL;DR

This paper proves global well-posedness and scattering for radial solutions of a critical wave equation with a cubic convolution in dimensions four and higher, using concentration compactness and virial analysis to exclude soliton and cascade scenarios.

Contribution

It establishes the first global well-posedness and scattering result for this specific cubic convolution wave equation in higher dimensions.

Findings

Radial solutions are global and scatter under the given conditions.

The soliton-like and cascade scenarios are ruled out via energy and regularity arguments.

The proof employs concentration compactness, Duhamel formulas, and virial analysis.

Abstract

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution $\partial_{t}^2u-\Delta u=\pm(|x|^{-3}\ast|u|^2)u$ 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.