Interpolation scattering for wave equations with singular potentials and singular data

TL;DR

This paper develops a scattering theory for wave equations with singular potentials in weak-$L^p$ spaces, establishing global well-posedness, scattering, and decay properties using dispersive estimates and fixed point methods.

Contribution

It introduces a novel framework for scattering in weak-$L^p$ spaces for wave equations with singular potentials, combining dispersive and Lorentz space techniques.

Findings

Proves global well-posedness in weak-$L^p$ spaces.

Establishes scattering results in the singular potential framework.

Improves decay estimates for wave solutions in weak-$L^p$ spaces.

Abstract

In this paper we investigate a construction of scattering for wave-type equations with singular potentials on the whole space $\mathbb{R}^n$ in a framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for wave groups on Lorentz spaces and fixed point arguments to prove the global well-posedness for wave-type equations on weak-$L^p$ spaces. Then, we provide a corresponding scattering results in such singular framework. Finally, we use also the dispersive estimates to establish the polynomial stability and improve the decay of scattering in weak-$L^p$ spaces.