Small-data scattering for nonlinear waves with potential and initial data of critical decay
Paschalis Karageorgis
TL;DR
The article advances scattering theory for nonlinear wave equations with a short-range potential by addressing the critical decay regime for small data. It introduces a weighted functional framework and derives detailed a priori estimates for the Duhamel contributions of both the nonlinearity and the potential, enabling a fixed-point construction of global solutions in the radially symmetric setting. The main results establish the existence of a scattering operator at critical decay and show asymptotic freeness, with energy-decay estimates that quantify convergence to free states as $t\to\pm\infty$. These findings extend previously known results for $V\equiv0$ and for supercritical decay to the critical case with a decaying potential, highlighting the interplay between nonlinearity, dimension, and short-range perturbations in wave scattering. The methods are framed in a robust weighted-energy approach that could inform further studies of critical nonlinear wave phenomena with potentials.
Abstract
We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp existence results for the Cauchy problem with small initial data; those require the data to decay at a rate greater than or equal to a critical decay rate which depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case. In this paper, we extend the scattering results to the critical case and we also allow the presence of a short-range potential.