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

### Changxing Miao, Jianwei Yang and Tengfei Zhao

Vol. 305 (2020), No. 1, 251–290
##### Abstract

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 ${Ḃ}_{1,1}^{3}×{Ḃ}_{1,1}^{2}\left({ℝ}^{5}\right)$. This is the $5$-dimensional analogue of Dodson’s result (2019), which was the first 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 Dodson’s strategy (2019), where we also avoid a logarithm-type loss by employing the inhomogeneous Strichartz estimates.

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##### Keywords
nonlinear wave equation, Strichartz estimates, scattering, hyperbolic coordinates, Morawetz estimates
##### Mathematical Subject Classification 2010
Primary: 35B40, 35L05
Secondary: 35Q40
##### Milestones
Received: 25 June 2018
Accepted: 31 October 2019
Published: 17 March 2020
##### Authors
 Changxing Miao Institute of Applied Physics and Computational Mathematics Beijing China Jianwei Yang Beijing Institute of Technology Beijing China Tengfei Zhao Beijing Computational Science Research Center Beijing China