Scattering for Wave Equations with Sources and Slowly Decaying Data

Explore a 43-minute conference talk on scattering for wave equations with sources and slowly decaying data, presented by Hans Lindblad at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the construction of solutions with prescribed radiation fields for wave equations featuring polynomially decaying sources near the lightcone. Examine the motivation behind this research, stemming from semilinear wave equations satisfying the weak null condition. Discover how solutions to the forward problem exhibit a logarithmic leading order term on the lightcone and non-trivial homogeneous asymptotics within the lightcone's interior. Investigate the backward scattering solutions constructed from source knowledge and radiation field at null infinity, focusing on their second-order explicit asymptotic solutions and novel matching conditions near the lightcone. Gain insights into the meticulous analysis of forward solutions with sources on the lightcone. Explore the relationship between radiation field asymptotics towards space-like infinity and explicit homogeneous solutions outside the lightcone for slowly polynomially decaying data, corresponding to mass, charge, and angular momentum in applications. Uncover the surprising discovery that these data can generate the same logarithmic radiation field as the source term, necessitating a thorough analysis of the forward homogeneous solution near the lightcone using Funk transform invertibility. Learn about this collaborative work with Volker Schlue, presented as part of the Thematic Programme on "Nonlinear Waves and Relativity" at ESI.