Time-weighted estimates in Lorentz spaces and self-similarity for wave equations with singular potentials

Abstract

We show time-weighted estimates in Lorentz spaces for the linear wave equation with singular potential. As a consequence, assuming radial symmetry on initial data and potentials, we obtain well-posedness of global solutions in critical weak-$L^p$ spaces for semilinear wave equations. In particular, we can consider the Hardy potential $V\left(x\right)=c|x{|}^{-2}$ for small $\left|c\right|$. Self-similar solutions are obtained for potentials and initial data with the right homogeneity. Our approach relies on performing estimates in the predual of weak-$L^p$, i.e., the Lorentz space $L^{\left(p^{\prime },1\right)}$.