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2003, Volume 9, Issue 6: 1387-1400. Doi: 10.3934/dcds.2003.9.1387
This issue Previous Article Sustainable dynamical systems Next Article Diophantine conditions in small divisors and transcendental number theory

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

  • 1.

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

  • 2.

    Department of Mathematics, Princeton University, Princeton N.J. 08544

  • 3.

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

Received:  December 31, 2001
Revised:  February 28, 2003
Published:  October 2003
Abstract Related Papers Cited by
    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.
    Mathematics Subject Classification: 35L05, 35L15.

    Citation:
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