American Institute of Mathematical Sciences

Advanced
March  2008, 1(1): 71-82. doi: 10.3934/dcdss.2008.1.71

On radial solutions of semi-relativistic Hartree equations

Yonggeun Cho 1, and  Tohru Ozawa 2,

1. 

Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea
2. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810

Received  July 2006 Revised  August 2007 Published  December 2007

We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 < \gamma < n, n \ge 1$. In [2, 3], the global well-posedness (GWP) was shown for the value of $\gamma \in (0, \frac{2n}{n+1}), n \ge 2$ with large data and $\gamma \in (2, n), n \ge 3$ with small data. In this paper, we extend the previous GWP result to the case for $\gamma \in (1, \frac{2n-1}n), n \ge 2$ with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.
Keywords: radially symmetric solution., global well-posedness, semi-relativistic Hartree type equation.
Mathematics Subject Classification: Primary: 35Q40, 35Q55; Secondary: 47J3.
Citation: Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71
[1]

Qingxuan Wang, Binhua Feng, Yuan Li, Qihong Shi. On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022017
[2]

Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725
[3]

Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047
[4]

Kiyeon Lee. Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3683-3702. doi: 10.3934/cpaa.2021126
[5]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053
[6]

Huy Tuan Nguyen, Nguyen Anh Tuan, Chao Yang. Global well-posedness for fractional Sobolev-Galpern type equations. Discrete & Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2021206
[7]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
[8]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[9]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[10]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[11]

Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483
[12]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052
[13]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[14]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[15]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273
[16]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[17]

Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
[18]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[19]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[20]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy