Scattering from infinity for semilinear wave equations satisfying the null condition or the weak null condition

TL;DR

This paper proves global existence for semilinear wave equations satisfying null or weak null conditions, using advanced asymptotic analysis and novel estimates, with results applicable to physically relevant equations.

Contribution

It introduces a new fractional Morawetz estimate and higher order asymptotic expansion to establish sharp global existence results for equations with weak null conditions.

Findings

Global existence backwards from scattering data at infinity.

Sharp decay rates matching radiation field behavior.

Handling of logarithmic terms in weak null condition cases.

Abstract

We show global existence backwards from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behaviour that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.