De Almeida, Marcelo F.; Ferreira, Lucas C. F. Time-weighted estimates in Lorentz spaces and self-similarity for wave equations with singular potentials. (English) Zbl 1361.35037 Anal. PDE 10, No. 2, 423-438 (2017). Summary: 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 \(V(x)=c| x|^{-2}\) 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-\(L^{p}\), i.e., the Lorentz space \(L^{(p',1)}\). Cited in 1 Document MSC: 35C06 Self-similar solutions to PDEs 35L05 Wave equation 35L71 Second-order semilinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B06 Symmetries, invariants, etc. in context of PDEs 42B35 Function spaces arising in harmonic analysis Keywords:radial symmetry; Lorentz spaces; critical weak-\(L^{p}\) spaces; predual of weak-\(L^{p}\) PDFBibTeX XMLCite \textit{M. F. De Almeida} and \textit{L. C. F. Ferreira}, Anal. PDE 10, No. 2, 423--438 (2017; Zbl 1361.35037) Full Text: DOI