Vol. 10, No. 2, 2017

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

Marcelo F. de Almeida and Lucas C. F. Ferreira

Vol. 10 (2017), No. 2, 423–438
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 $|c|$. 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 ${L}^{\left({p}^{\prime },1\right)}$.

Keywords
wave equations, singular potentials, self-similarity, radial symmetry, Lorentz spaces
Mathematical Subject Classification 2010
Primary: 35L05, 35L71, 35L15, 35A01, 35B06
Secondary: 35C06, 42B35
Milestones
Received: 2 July 2016
Accepted: 12 November 2016
Published: 23 February 2017
Authors
 Marcelo F. de Almeida Departamento de Matemática Universidade Federal de Sergipe, CCET Av. Marechal Rondon, s/n 49100-000 Aracaju-SE Brazil Lucas C. F. Ferreira Departamento de Matemática Universidade Estadual de Campinas, IMECC Rua Sérgio Buarque de Holanda, 651 13083-859 Campinas-SP Brazil