DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SCATTERING THEORY FOR A WAVE EQUATION
OF HARTREE TYPE
KIMITOSHI TSUTAYA
We consider a scattering problem for a wave equation with a cubic convolution together
with a potential in three space dimensions. We show the sharp conditions on the decay
rates at infinity of the potentials and initial data for the existence of scattering operators.
Copyright © 2006 Kimitoshi Tsutaya. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
This paper is intended to present an extension of the result in the paper [9], which treats
the Cauchy problem.
We consider a scattering problem for the nonlinear wave equation
∂ 2 t u − Δu = V 1 (x)u + ( V 2 ∗ u 2) u in R × R 3 , (NW)
where V 1 (x) = O(|x| −γ1 )as|x|→∞, V 2 (x) = ν 2 |x| −γ2 , ν 2 ∈ R, γ 1 ,γ 2 > 0, and ∗ denotes
spatial convolution. The potential V 1 is assumed to be a smooth function.
The Schrödinger equation with the interaction term V 1 (x)u +(V 2 ∗ u 2 )u was studied
by Hayashi and Ozawa [3]. See also Coclite and Georgiev [2].
We study the scattering problem for (NW) with small initial data. Moreover, in this
paper the potential V 1 is assumed to be small since the solution may blow up in a finite
time unless V 1 is small. See Strauss and Tsutaya [8].
In case V 1 (x) ≡ 0, the initial data have small amplitude, and V 2 (x) satisfies some conditions,
it is known by [4–7] that the scattering operator exists for small initial data.
In particular, Mochizuki and Motai [7] proved the existence of scattering operators for
(NW)inn-dimensional space if
2
2+
3(n − 1)