Consider the nonlinear wave equation with zero mass in two space dimensions. When it comes to the associated Cauchy problem with small initial data, the known existence results are already sharp; those require the data to decay at a rate $k\geq k_c$, where $k_c$ is a critical decay rate that depends on the order of the nonlinearity. However, the known scattering results treat only the supercritical case $k>k_c$. In this paper, we prove the existence of the scattering operator for the full optimal range $k\geq k_c$.
"Small-data scattering for nonlinear waves of critical decay in two space dimensions." Differential Integral Equations 19 (6) 601 - 626, 2006. https://doi.org/10.57262/die/1356050355