Remarks on scattering theory and large time asymptotics of solutions to hartree type equations with a long range potential

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)

Abstract

We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations { iu t = - 1/2Δu + f(|u| 2u, (t, x) ∈ R × R n, u(0,x) = u 0(x), x ∈ R n, n ≥ 2, where the nonlinear interaction term is f(|u| 2) = V * |u| 2, V(x) = λ|x| - δ λ ∈ R,0 < δ <1. We suppose that the initial data u 0 ∈ H 0.l and the value ∈= ||u 0||H 0.1 is sufficiently small, where l is an integer satisfying l ≥ [n/2] + 3, and [s] denotes the largest integer less than s. Then we prove that there exists a unique final state u + ∈ H 0,l-2 such that for all t > 1 u(t, x) = 1 û/(it) n/2 + (x/t) exp(ix 2/2t-it 1- δ/1- δ f(|û+| 2)(x/t) + 0(1+t 1-2δ))+0( -n/2-δ) uniformly with respect to x ∈ R n with the following decay estimate ||u(t) ||L p ≤ C∈t n/p-n/2, for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ <1 there exists a unique final state u + ∈ H 0,1-2 such that for all t ≥ 1, ||u(t)-exp(-it 1-δ/1- δ f(|û+| 2(x/t)) U(t)u+||L 2 = 0(t 1-2 δ), and uniformly with respect to x ∈ R n u(t, x) = 1/(it) n/2 û+(x/t)exp(ix 2/2t-it 1- δ/1- δ f(|û +

Original languageEnglish
Pages (from-to)13-24
Number of pages12
JournalSUT Journal of Mathematics
Volume34
Issue number1
Publication statusPublished - 1998 Dec 1
Externally publishedYes

Keywords

  • Asymptotics in time
  • Hartree type equations
  • Long range potential

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Remarks on scattering theory and large time asymptotics of solutions to hartree type equations with a long range potential'. Together they form a unique fingerprint.

Cite this