Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

Abstract

In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in [33]. Starting from a global solution u to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system. 2020 Mathematics Subject Classification. 35L70.

Nontrivial global solutions to some quasilinear wave equations in three space dimensions

“…The method used in the current paper is essentially the same as that in the author's recent paper [53] or Chapters 2 and 3 in the author's PhD dissertation [52] on the modified wave operators for a scalar quasilinear wave equation. To construct a global solution, we solve the Cauchy problem (1.1) backwards from infinite time.…”

“…That is, starting with a certain asymptotic profile, we find a global solution to (1.1) that matches this asymptotic profile at infinite time. The asymptotic profile is given by a global solution (µ, U)(s, q, ω) to the geometric reduced system introduced in [52,53]. Here we are not able to solve the geometric reduced system to a general system (1.1) explicitly, so we start with a given global solution (µ, U) for s ≥ 0 which satisfies several pointwise estimates; see Section 1.3 below.…”

“…The main tool to construct global solutions in this paper is the geometric reduced system introduced by the author. In [53], this reduced system is derived in the scalar case, and it is extended to the vector-valued case in Chapter 2 of the author's PhD dissertation [52]. Our analysis starts as in Hörmander's derivation in [15][16][17], but diverges at a key point: the choice of q is different.…”

A uniqueness theorem for 3D semilinear wave equations satisfying the null condition

“…In this case, we construct an asymptotic profile, not by taking the limit (1.12), but by solving a certain system of asymptotic equations. For example, we can take our asymptotic profile as a solution to the Hörmander's asymptotic equation (1.9), or to the geometric reduced system introduced by the author [41].…”

“…The first one is called asymptotic completeness and the second one is called existence of (modified) wave operators. We refer our readers to [6,7,32,40,41] for some work on (modified) scattering theory for nonlinear wave equations.…”