Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Dispersive estimate for the wave equation with the inverse-square potential

Pages: 1387 - 1400, Volume 9, Issue 6, November 2003      doi:10.3934/dcds.2003.9.1387

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Fabrice Planchon - Laboratoire Analyse, Géométrie & Applications, UMR 7539, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France (email)
John G. Stalker - Department of Mathematics, Princeton University, Princeton N.J. 08544, United States (email)
A. Shadi Tahvildar-Zadeh - Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway NJ 08854, United States (email)

Abstract: 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:  Wave equation, inverse-square potential, dispersive estimates, time decay.
Mathematics Subject Classification:  35L05, 35L15.

Received: January 2002;      Revised: March 2003;      Available Online: September 2003.