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

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 ${\dot{B}}_{1,1}^3\times {\dot{B}}_{1,1}^2({\mathbb{R}}^5)$. 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.