Small analytic solutions to the Hartree equation

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• 作者：Hironobu Sasaki¹ ; ^{sasaki@math.s.chiba-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q55 ; 35B65 ; 35G25
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：270
• 期：3
• 页码：1064-1090
• 全文大小：429 K

文摘

We consider the Cauchy problem for the Hartree equation in space dimension d≥3$d\ge 3$. We assume that the interaction potential V   belongs to the weak L^d/2$L^{d/2}$ space. We prove that if the initial data ϕ   is sufficiently small in the L²$L^2$-sense and either ϕ   or its Fourier transform Fϕ$\mathcal{F}\varphi$ satisfies a real-analytic condition, then the solution 560b765d1e84fcdd3f671fa673" title="Click to view the MathML source">u(t)$u(t)$ is also real-analytic for any t≠0$t\ne 0$. We also prove that if ϕ and V   satisfy some strong condition, then 560b765d1e84fcdd3f671fa673" title="Click to view the MathML source">u(t)$u(t)$ can be extended to an entire function on 35b84967045809c9ad6aad" title="Click to view the MathML source">C^d${\mathbb{C}}^d$ for any t≠0$t\ne 0$. A part of our method is applicable to the final value problem. We remark that no L²$L^2$ smallness condition is imposed on first and higher order partial derivatives of ϕ   and Fϕ$\mathcal{F}\varphi$.