Propagation of Singularities for Semi-Linear Wave Equations With Nonlinearity Satisfying the Null Condition

Abstract

We study the propagation of singularities for semi-linear wave equations u = F (u, Du) satisfying the null condition, and improve an earlier result of ours. Microlocal Sobolev regularity in H s (U ) ∩ H r ml propagates along null bicharacteristics, when a suitable condition is imposed on s and r. We provide here optimal conditions on s and r, and improve existing results to include values up to n/2 < s r < 3s − n when the nonlinearity F (u, Du) satisfies the null condition.

Scattering for wave equations with sources close to the lightcone and prescribed radiation fields

“…The system (1.34) is the simplest example of a system of wave equations satisfying the weak null condition, see [34], and is itself a model of the Einstein equations in harmonic coordinates [35,36,32]. We refer the reader to the introduction of [38], for a discussion of the asymptotics established in [32], and the relevant semi-linear model problems in this setting.…”

“…In [38] we approached the scattering problem for a class of semi-linear wave equations and showed the existence of solutions from scattering data at infinity, which was assumed to decay along null infinity. We showed that approximate solutions can be constructed for non-linear wave equations satisfying the null condition, and used a version of the fractional Morawetz estimate, see [41], to control the remainder for the backwards problem, and showed decay of the solution at a rate corresponding to the decay of the radiation field.…”

“…Since then this approach has been applied and extended in several directions: Lili He proved a scattering result for the Maxwell Klein Gordon-equations [18]. Solutions to this system have slower decay in the interior than those treated in [38] and to prove existence from infinity it is necessary to subtract the leading order homogeneous interior asymptotics. In fact, a suitable model for the weak null structure of this system is, as discussed in [18],…”

“…In settings where the asymptotics are captured by (1.2), and the decay in the interior is sufficiently fast, there is a well-defined scattering problem: Given a radiation field F 0 , show the existence of a unique solution to (1.1) so that (1.2) holds with the prescribed radiation field F 0 . For example, we have shown in [38] that for radiation fields that decay like |F 0 (q, ω)| 1/ q γ , with 1/2 < γ < 1, there exists a unique solution to (1.1), say with F = 0, so that (1.2) holds, and moreover the solutions decay globally, |φ| 1 t + r t − r γ , at a rate γ < γ.…”