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### Open Access Journals

DCDS

We prove that Strichartz-type $L^p$ estimates hold for solutions of
the linear wave equation with the inverse square potential, under the
additional assumption that the Cauchy data are spherically symmetric.
The estimates are then applied to prove global well-posedness in the critical
norm for a nonlinear wave equation.

DCDS

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 $L^\infty$ 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.

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