Dispersive Estimate for the Wave Equation

Fabrice Planchon

Description

Dispersive Estimate for the Wave Equation with the Inverse-Square Potential Fabrice Planchon John G. Stalker A. Shadi Tahvildar-Zadeh 12/27/01 Abstract We prove that spherically symmetric solutions of the Cauchy prob- lem for the linear wave equation with the inverse-square potential sat- isfy a modied dispersive inequality that bounds the L 1 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. 1 Introduction Consider the following linear wave equation 8 < : n u+ a jxj 2 u = 0 u(0; x) = f(x) @ t u(0; x) = g(x) (1.1) where n = @ 2 t n is the D'Alembertian in R n+1 and a is a real number. The interest in this equation comes from the potential term being homogeneous of degree -2 and therefore scaling the same way as the D'Alembertian term.

• references cited there


• wave equation


• dispersive estimate


• physical space via legendre


• multiplication operator


• inverse-square potential


• then obtain


• following dispersive


• bessel function

Sujets

• Planchon
• Wave equation
• Multiplication operator
• Bessel function

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

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erse-Square
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oten
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erturbation
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abrice
in
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hon
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John
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8
Stalk
x
er
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A.
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Shadi
g
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ah
b
vildar-Zadeh
is
12/27/01
in
Abstract
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or

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space
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follo
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is
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main
distinction
result
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of
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this
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pap
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er:
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THEOREM
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ph
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tegration
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o
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et
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and
b
decomp
e
op
the
ar
r
the
adial
w
solution
represen
of
solution
(1.1)