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November  2003, 9(6): 1387-1400. doi: 10.3934/dcds.2003.9.1387

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

Fabrice Planchon 1, , John G. Stalker 2, and  A. Shadi Tahvildar-Zadeh 3,

1. 

Laboratoire Analyse, Géométrie & Applications, UMR 7539, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France
2. 

Department of Mathematics, Princeton University, Princeton N.J. 08544, United States
3. 

Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway NJ 08854, United States

Received  January 2002 Revised  March 2003 Published  September 2003

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..
Mathematics Subject Classification: 35L05, 35L1.
Citation: Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
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