- Main
Modified scattering for a scalar quasilinear wave equation satisfying the weak null condition
- Yu, Dongxiao
- Advisor(s): Tataru, Daniel
Abstract
The objective of this dissertation is to study the long time dynamics of a scalar quasilinear wave equation
To study modified scattering, we first identify a notion of asymptotic profile and an associated notion of scattering data. One candidate for the asymptotic profile is given by the asymptotic PDE
In Chapter 3, we prove the existence of the modified wave operators for the scalar quasilinear wave equation. Fixing a scattering data which is the initial data for the geometric reduced system, we can first construct an approximate solution to the model equation. Then, by studying a backward Cauchy problem, we show that there exists a global solution to the scalar quasilinear wave equation which matches the approximate solution at infinite time.
In Chapter 4, we prove the asymptotic completeness for the same equation. Given a global solution to the scalar quasilinear wave equation, we rigorously derive the geometric reduced system with error terms. These allow us to recover the scattering data, as well as to construct a matching exact solution to the reduced system.
Main Content
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