The global well-posedness and scattering for the $5$D defocusing conformal invariant NLW with radial initial data in a critical Besov space

TL;DR

This paper proves global well-posedness and scattering for radial solutions of a 5D defocusing conformal invariant nonlinear wave equation with initial data in a critical Besov space, advancing understanding of such equations without relying on conservation laws.

Contribution

It extends prior results by establishing global solutions and scattering in a critical Besov space for the 5D conformal NLW without the need for uniform Sobolev norm bounds.

Findings

Proved global well-posedness for radial solutions in a critical Besov space.

Established scattering results for the 5D defocusing conformal NLW.

Developed Strichartz estimates tailored for radial solutions.

Abstract

In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space $\dot{B}^3_{1,1}\times\dot{B}^2_{1,1}(\mathbb{R}^5)$. This is the five dimensional analogue of \cite{dodson-2016}, which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invariance of the equation. The proof is based on exploiting the structure of the radial solution, developing the Strichartz-type estimates and incorporation of the strategy in \cite{dodson-2016}, where we also avoid a logarithm-type loss by employing the inhomogeneous Strichartz estimates.