On the global Cauchy problem for the Hartree equation with rapidly decaying initial data

Hyakuna, Ryosuke

doi:10.1016/j.anihpc.2018.11.004

Hyakuna, Ryosuke

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1081-1104.

Résumé

This paper is concerned with the Cauchy problem for the Hartree equation on $R^{n}, n \in N$ with the nonlinearity of type $(| \cdot |^{- γ} ⁎ | u |^{2}) u, 0 < γ < n$ . It is shown that a global solution with some twisted persistence property exists for data in the space $L^{p} \cap L^{2}, 1 \leq p \leq 2$ under some suitable conditions on γ and spatial dimension $n \in N$ . It is also shown that the global solution u has a smoothing effect in terms of spatial integrability in the sense that the map $t \mapsto u (t)$ is well defined and continuous from $R ∖ {0}$ to $L^{p^{'}}$ , which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat $L^{p}$ -spaces are also presented. The local and global results are proved by combining arguments by Carles–Mouzaoui with a new functional framework introduced by Zhou. Furthermore, it is also shown that the global results can be improved via generalized dispersive estimates in the case of one space dimension.

Zbl

DOI : 10.1016/j.anihpc.2018.11.004

Classification : 35Q55
Mots-clés : Nonlinear Schrödinger equations, Hartree equation, Cauchy problem, Global well-posedness, $ {L}^{p}$-Cauchy data, Rapidly decaying data

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     author = {Hyakuna, Ryosuke},
     title = {On the global {Cauchy} problem for the {Hartree} equation with rapidly decaying initial data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1081--1104},
     publisher = {Elsevier},
     volume = {36},
     number = {4},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.11.004},
     zbl = {1421.35337},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.004/}
}

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Hyakuna, Ryosuke. On the global Cauchy problem for the Hartree equation with rapidly decaying initial data. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1081-1104. doi : 10.1016/j.anihpc.2018.11.004. http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.004/

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