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January 1998 REMARKS ON SCATTERING THEORY AND LARGE TIME ASYMPTOTICS OF SOLUTIONS TO HARTREE TYPE EQUATIONS WITH A LONG RANGE POTENTIAL
Nakao Hayashi, Pavel I. Naumkin
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SUT J. Math. 34(1): 13-24 (January 1998). DOI: 10.55937/sut/991983736

Abstract

We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations

{iut=12Δu+f(|u|2)u,(t,x)R×Rn,u(0,x)=u0(x),xRn,n2,

where the nonlinear interaction term is f(|u|2)=V|u|2,V(x)=λ|x|δ, λR,0<δ<1. We suppose that the initial data u0H0,l and the value ϵ=u0H0,l is sufficiently small, where l is an integer satisfying l[n2]+3, and [s] denotes the largest integer less than s. Then we prove that there exists a unique final state u+H0,l2 such that for all t>1

u(t,x)=1(it)n2u^+(xt)exp(ix22tit1δ1δf(|u^+|2)(xt)+O(1+t12δ))+O(tn/2δ)

uniformly with respect to xRn with the following decay estimate u(t)LpCϵtnpn2, for all t1 and for every 2p. Furthermore we show that for 12<δ<1 there exists a unique final state u+H0,l2 such that for all t1

u(t)exp(it1δ1δf(|u^+|2)(xt))U(t)u+L2=o(t12δ)

and uniformly with respect to xRn

u(t,x)=1(it)n2u^+(xt)exp(ix22tit1δ1δf(|u^+|2)(xt))+O(tn/2+12δ),

where ϕ^ denotes the Fourier transform of the function ϕ,Hm,s={ϕS:ϕm,s=(1+|x|2)s/2(1Δ)m/2ϕL2<},m,sR. In [5] we assumed that u0Hm,0H0,m,(m=n+2), and showed the same results as in this paper. Here we show that we do not need regularity conditions on the initial data by showing the local existence theorem in lower order Sobolev spaces.

Citation

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Nakao Hayashi. Pavel I. Naumkin. "REMARKS ON SCATTERING THEORY AND LARGE TIME ASYMPTOTICS OF SOLUTIONS TO HARTREE TYPE EQUATIONS WITH A LONG RANGE POTENTIAL." SUT J. Math. 34 (1) 13 - 24, January 1998. https://doi.org/10.55937/sut/991983736

Information

Received: 12 September 1997; Published: January 1998
First available in Project Euclid: 18 June 2022

Digital Object Identifier: 10.55937/sut/991983736

Subjects:
Primary: 35Q55

Keywords: Asymptotics in time , Hartree type equations , long range potential

Rights: Copyright © 1998 Tokyo University of Science

Vol.34 • No. 1 • January 1998
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