Geodesic
Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e108

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We study the timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Given a global solution u to the scalar wave equation with sufficiently small $C_c^\infty $ initial data, we derive an asymptotic formula for this global solution inside the light cone (i.e. for $|x|). It involves the scattering data obtained in the author’s asymptotic completeness result in [75]. Using this asymptotic formula, we prove that u must vanish under some decaying assumptions on u or its scattering data, provided that the wave equation violates the null condition. 
Yu, Dongxiao. Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e108. doi: 10.1017/fms.2025.10072
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