Interpolation scattering for wave equations with singular potentials and singular data

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.

💡 Research Summary

The paper studies the Cauchy problem for a semilinear wave equation on the whole Euclidean space $\mathbb{R}^{n}$ with $n\ge5$, where the linear part contains a Hardy‑type singular potential $V_{1}(x)=c_{1}|x|^{-2}$ and the nonlinear part is multiplied by a second singular weight $V_{2}(x)=c_{2}|x|^{b}$ with $0<b<2$. The nonlinearity $F$ satisfies a standard Lipschitz‑type condition of order $q>1$:
\