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-Lp spaces for semilinear wave equations. In particular, we can consider the Hardy potential V (x) = c|x|−2 for small |c|. 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-Lp, i.e., the Lorentz space L(p′,1) .