We prove that spherically symmetric solutions of the Cauchy problem for the linear wave equation with the inverse-square potential satisfy a modified dispersive inequality that bounds the L∞ norm of the solution in terms of certain Besov norms of the data, with a factor that decays in t for positive potentials. When the potential is negative we show that the decay is split between t and r, and the estimate blows up at r = 0. We also provide a counterexample showing that the use of Besov norms in dispersive inequalities for the wave equation are in general unavoidable.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Dispersive estimates
- Inverse-square potential
- Time decay
- Wave equation