For 2 〈 y 〈 min{4, n}, we consider the focusing Hartree equation iut ＋ Au ＋ （｜x｜^-γ ＊ ｜u｜2）u = O, x∈ R^n Let M[u] and E[u] denote the mass and energy, respectively, of a solution u, and Q be the ground st...For 2 〈 y 〈 min{4, n}, we consider the focusing Hartree equation iut ＋ Au ＋ （｜x｜^-γ ＊ ｜u｜2）u = O, x∈ R^n Let M[u] and E[u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of - △ ＋ Q = （｜x｜^-γ ＊ ｜Q｜^2）Q. Guo and Wang [Z. Angew. Math. Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of （0,1） if M[u]^l-ScE[u]^Sc 〈 M[Q] ^1-sc E[Q] ^（sc= r-2/2）. In this paper, we consider the complementary case M[u]^1-ScE[u]^sc 〉_ M[Q]^1-sc and obtain a criteria on blow-up and global existence for the Hartree equation （0.1）.展开更多
We consider the scattering problem for the Hartree equation with potential|x|<sup>-1</sup>in a space of dimension n≥2.We prove the existence of H<sup>m</sup>-modified wave operator for Hartree...We consider the scattering problem for the Hartree equation with potential|x|<sup>-1</sup>in a space of dimension n≥2.We prove the existence of H<sup>m</sup>-modified wave operator for Hartree equation on a dense set of a neighborhood of zero in H<sup>m</sup>（R<sup>n</sup>）,meanwhile,we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimension n≥2.展开更多
This article considers a family of 3D-Hartree-type equation with Coulomb potential ｜x｜^-1, whose initial data oscillates so that a caustic appears. In the linear geometric optics case, by using the Lagrangian integr...This article considers a family of 3D-Hartree-type equation with Coulomb potential ｜x｜^-1, whose initial data oscillates so that a caustic appears. In the linear geometric optics case, by using the Lagrangian integrals, a uniform description of the solution outside the caustic, and near the caustic are obtained.展开更多
The instability property of the standing wave uω（t, x） = eiωtφ（x） for the Klein–Gordon– Hartree equation is investigated. For the case N≥3 and w2 〈2/N＋4-γ,it is shown that the standing wave eiwtφ（x） is...The instability property of the standing wave uω（t, x） = eiωtφ（x） for the Klein–Gordon– Hartree equation is investigated. For the case N≥3 and w2 〈2/N＋4-γ,it is shown that the standing wave eiwtφ（x） is strongly unstable by blow-up in finite time.展开更多
In this article, we give a simple proof on the energy scattering for the Hartree equations using the interaction Morawetz estimate that was originally introduced in [5].
基金supported by the National Natural Science Foundation of China(11371267)Sichuan Province Science Foundation for Youths(2012JQ0011)
文摘For 2 〈 y 〈 min{4, n}, we consider the focusing Hartree equation iut ＋ Au ＋ （｜x｜^-γ ＊ ｜u｜2）u = O, x∈ R^n Let M[u] and E[u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of - △ ＋ Q = （｜x｜^-γ ＊ ｜Q｜^2）Q. Guo and Wang [Z. Angew. Math. Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of （0,1） if M[u]^l-ScE[u]^Sc 〈 M[Q] ^1-sc E[Q] ^（sc= r-2/2）. In this paper, we consider the complementary case M[u]^1-ScE[u]^sc 〉_ M[Q]^1-sc and obtain a criteria on blow-up and global existence for the Hartree equation （0.1）.
基金This project is supported by the National Natural Science Foundation of China,19601005
文摘We consider the scattering problem for the Hartree equation with potential|x|<sup>-1</sup>in a space of dimension n≥2.We prove the existence of H<sup>m</sup>-modified wave operator for Hartree equation on a dense set of a neighborhood of zero in H<sup>m</sup>（R<sup>n</sup>）,meanwhile,we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimension n≥2.
文摘This article considers a family of 3D-Hartree-type equation with Coulomb potential ｜x｜^-1, whose initial data oscillates so that a caustic appears. In the linear geometric optics case, by using the Lagrangian integrals, a uniform description of the solution outside the caustic, and near the caustic are obtained.
基金The first author is supported by the Key Project of Chinese Ministry of Education(Grant No.211162)Sichuan Province Science Foundation for Youths(Grant No.2012JQ0011)the second author is supported byNational Natural Science Foundation of China(Grant No.11371267)
文摘The instability property of the standing wave uω（t, x） = eiωtφ（x） for the Klein–Gordon– Hartree equation is investigated. For the case N≥3 and w2 〈2/N＋4-γ,it is shown that the standing wave eiwtφ（x） is strongly unstable by blow-up in finite time.
基金supported by the NSF of China (10801015, 10901012)
文摘In this article, we give a simple proof on the energy scattering for the Hartree equations using the interaction Morawetz estimate that was originally introduced in [5].
