Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition (Q6605399)

In this paper, the author proves the asymptotic completeness result for a scalar quasilinear wave equation, \(g^{\alpha\beta}(u)\partial_\alpha \partial_\beta u=0\), in \(\mathbb{R}^{1+3}_{t,x}\), which satisfies the weak null condition. As a model equation which does not satisfy the null condition but admits global solutions, for any sufficiently small, compactly supported initial data, this equation has been extensively investigated. It is known from the works of \textit{H. Lindblad} [Am. J. Math. 130, No. 1, 115--157 (2008; Zbl 1144.35035)] and \textit{S. Alinhac} [Astérisque 284, 1--91 (2003; Zbl 1053.35097)], that the equation admits global solutions which do not behave like linear solutions, in the sense they blow up at infinity. Then the long time dynamics for the solutions becomes a central problem for this equation. In [Commun. Pure Appl. Math. 73, No. 5, 1035--1099 (2020; Zbl 1445.35252)], \textit{Y. Deng} and \textit{F. Pusateri} gave a detailed description of the behavior of solutions close to the light cone, including the blow-up at infinity. Recently, in [Commun. Math. Phys. 382, No. 3, 1961--2013 (2021; Zbl 1461.35159)], the author proved the existence of the modified wave operators, by using a new reduced system. In this work, the author continues the study of the modified scattering theory, by proving an asymptotic completeness result. Starting from a global solution \(u\) to the quasilinear wave equation, certain well-chosen asymptotic variables are proved to solve the same reduced system with small error terms. This allows the author to recover the scattering data for the system, as well as to construct a matching exact solution to the reduced system.