Scattering theory for the wave equation of a Hartree type in three space dimensions

Kimitoshi Tsutaya

doi:10.3934/dcds.2014.34.2261

Article Contents

2014, Volume 34, Issue 5: 2261-2281. Doi: 10.3934/dcds.2014.34.2261

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Scattering theory for the wave equation of a Hartree type in three space dimensions

Kimitoshi Tsutaya^1,

1.
Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561

Received: December 2012

Revised: August 2013

Published: May 2014

Abstract / Introduction Related Papers Cited by

Abstract

The paper concerns a scattering problem of the wave equation of a Hartree type with small initial data with fast decay. The equation is \[ \partial_t^2 u - \Delta u = V_1(x)u+ (V_2\ast |u|^{p-1})u , \qquad t\in {\bf R}, \; x \in {\bf R}^3, \] where $p\ge 3, \; V_1(x)=O(|x|^{-\gamma_1})$ with $\gamma_1>0$ as $|x|\to\infty, \; V_2(x) = \pm |x|^{-\gamma_2}$ with $\gamma_2>0$. We prove the existence of scattering operators under almost optimal conditions on the potentials and initial data in terms of decay, using pointwise estimates. Our result generalizes the one by [14, 15] for the case $p=3$.

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Mathematics Subject Classification: Primary: 35L05, 35L70; Secondary: 35P25.

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References

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