DCDS
$L^p$ Estimates for the wave equation with the inverse-square potential
Fabrice Planchon John G. Stalker A. Shadi Tahvildar-Zadeh
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 427-442 doi: 10.3934/dcds.2003.9.427
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.
keywords: inverse-square potential conjugation operator. space-time estimates Wave equation
DCDS
Dispersive estimate for the wave equation with the inverse-square potential
Fabrice Planchon John G. Stalker A. Shadi Tahvildar-Zadeh
Discrete & Continuous Dynamical Systems - A 2003, 9(6): 1387-1400 doi: 10.3934/dcds.2003.9.1387
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.
keywords: inverse-square potential dispersive estimates Wave equation time decay.

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