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Time-weighted estimates in Lorentz spaces and self-similarity for wave equations with singular potentials. (English) Zbl 1361.35037

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 \(L^{(p',1)}\).

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
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