American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing
  • CPAA Home
  • This Issue
  • Next Article
    On analyticity up to the boundary for critical quasi-geostrophic equation in the half space
April  2022, 21(4): 1225-1247. doi: 10.3934/cpaa.2022017

On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

Qingxuan Wang 1,, , Binhua Feng 2, , Yuan Li 3, and  Qihong Shi 4,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China
2. 

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China
3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China
4. 

Department of Mathematics, Lanzhou University of Technology, Lanzhou, 730000, China

*Corresponding author

Received  September 2021 Revised  December 2021 Published  April 2022 Early access  December 2021

Fund Project: This work was jointly supported by the National Natural Science Foundation of China (11801519, 11971436, 12061040, 11701244), the Zhejiang Provincial Natural Science Foundation of China (LZ22A010001), the Outstanding Youth Science Fund of Gansu Province (20JR10RA111), the Natural Science Foundation of Gansu Province (21JR7RA150, 20JR5RA460)

We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by
$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $
where
$ 0<\alpha<1 $
 and
$ \beta>0 $
. Firstly we study the existence and stability of the maximal ground state
$ \psi_\beta $
 at
$ N = N_c $
, where
$ N_c $
 is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states
$ \psi_\beta $
 when
$ \beta\rightarrow 0^+ $
, and the optimal blow-up rate with respect to
$ \beta $
 will be calculated.
Keywords: Semi-relativistic Hartree Equation, Maximal ground states, combined Hartree-type nonlinearities, Asymptotic Properties.
Mathematics Subject Classification: Primary: 35B40, 35J20; Secondary: 35S10.
Citation: Qingxuan Wang, Binhua Feng, Yuan Li, Qihong Shi. On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1225-1247. doi: 10.3934/cpaa.2022017
References:
[1]

S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinb. A, 145 (2015), 73-90.  doi: 10.1017/S0308210513000450.

[2]

Y. Cho and T. Ozawa, On the semi-relativistic Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.

[3]

Y. Deng, Y. Guo and L. Lu, On the collapse and concentration of Bose-Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differ. Equ., 54 (2015), 99-118.  doi: 10.1007/s00526-014-0779-9.

[4]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[5]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220.  doi: 10.1007/s00028-017-0397-z.

[6]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[7]

J. Fröhlich, B. Lars, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.

[8]

J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.  doi: 10.1002/cpa.20186.

[9]

Y. Guo and R. Seiringer, On the Mass concentration for Bose-Einstein condensation with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.

[10]

Y. Guo and X. Zeng, Ground states of pseudo-relativistic boson stars under the critical stellar mass, Ann. I. H. Poincaré, 34 (2017), 1611-1632.  doi: 10.1016/j.anihpc.2017.04.001.

[11]

Y. Guo, X. Zeng and H. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Pioncaré, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.

[12]

S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137.  doi: 10.1016/j.na.2013.11.023.

[13]

D. Holm and V. Putkaradze, Formation of clumps and patches in selfaggregation of finite-size particles, Phys. D, 220 (2006), 183-196.  doi: 10.1016/j.physd.2006.07.010.

[14]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[15]

E. Lenzmann, Uniqueness of ground states for pseudo-relativistic Hartree equations, Anal. Partial Differ. Equ., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[16]

E. Lenzmann and M. Lewin, On singularity formation for the $L^2$-critical Boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.

[17]

E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics., Commun. Math. Phys., 112 (1987), 147-174. 

[18]

E. H. Lieb and M. Loss, Analysis 2ed. Grad. Stud. Math., Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.

[19]

X. Luo, Normalized standing waves for the Hartree equations, J. Differ. Equ., 267 (2019), 4493-4524.  doi: 10.1016/j.jde.2019.05.009.

[20]

A. Michelangeli and B. Schlein, Dynamical collapse of boson stars, Commun. Math. Phys., 311 (2012), 645-687.  doi: 10.1007/s00220-011-1341-7.

[21]

D. T. Nguyen, On Blow-up Profile of Ground States of Boson Stars with External Potential, J. Stat. Phys., 169 (2017), 395-422.  doi: 10.1007/s10955-017-1872-1.

[22]

F. Pusateri, Modified Scattering for the Boson Star Equation, Commun. Math. Phys., 332 (2014), 1203-1234.  doi: 10.1007/s00220-014-2094-x.

[23]

Q. Shi and C. Peng, Well-posedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonl. Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.

[24]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differ. Equ., 269 (2020), 6941-6987.  doi: 10.1016/j.jde.2020.05.016.

[25]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 1-43.  doi: 10.1016/j.jfa.2020.108610.

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[27]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Partial Differ. Equ., 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.

[28]

C. Topaz, A. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[29]

Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differ. Equ., 262 (2017), 2684-2704.  doi: 10.1016/j.jde.2016.11.004.

[30]

J. Yang and J. Yang, Existence and mass concentration of pseudo-relativistic Hartree equation, J. Math. Phys., 58 (2017), 1-22.  doi: 10.1063/1.4996576.

[31]

V. C. Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Ibero., 29 (2013), 1421-1436.  doi: 10.4171/RMI/763.

[32]

X. Zeng and L. Zhang, Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials, J. Math. Anal. Appl., 452 (2017), 47-61.  doi: 10.1016/j.jmaa.2017.02.053.

show all references

References:
[1]

S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinb. A, 145 (2015), 73-90.  doi: 10.1017/S0308210513000450.

[2]

Y. Cho and T. Ozawa, On the semi-relativistic Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.

[3]

Y. Deng, Y. Guo and L. Lu, On the collapse and concentration of Bose-Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differ. Equ., 54 (2015), 99-118.  doi: 10.1007/s00526-014-0779-9.

[4]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[5]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220.  doi: 10.1007/s00028-017-0397-z.

[6]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[7]

J. Fröhlich, B. Lars, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.

[8]

J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.  doi: 10.1002/cpa.20186.

[9]

Y. Guo and R. Seiringer, On the Mass concentration for Bose-Einstein condensation with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.

[10]

Y. Guo and X. Zeng, Ground states of pseudo-relativistic boson stars under the critical stellar mass, Ann. I. H. Poincaré, 34 (2017), 1611-1632.  doi: 10.1016/j.anihpc.2017.04.001.

[11]

Y. Guo, X. Zeng and H. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Pioncaré, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.

[12]

S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137.  doi: 10.1016/j.na.2013.11.023.

[13]

D. Holm and V. Putkaradze, Formation of clumps and patches in selfaggregation of finite-size particles, Phys. D, 220 (2006), 183-196.  doi: 10.1016/j.physd.2006.07.010.

[14]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[15]

E. Lenzmann, Uniqueness of ground states for pseudo-relativistic Hartree equations, Anal. Partial Differ. Equ., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[16]

E. Lenzmann and M. Lewin, On singularity formation for the $L^2$-critical Boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.

[17]

E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics., Commun. Math. Phys., 112 (1987), 147-174. 

[18]

E. H. Lieb and M. Loss, Analysis 2ed. Grad. Stud. Math., Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.

[19]

X. Luo, Normalized standing waves for the Hartree equations, J. Differ. Equ., 267 (2019), 4493-4524.  doi: 10.1016/j.jde.2019.05.009.

[20]

A. Michelangeli and B. Schlein, Dynamical collapse of boson stars, Commun. Math. Phys., 311 (2012), 645-687.  doi: 10.1007/s00220-011-1341-7.

[21]

D. T. Nguyen, On Blow-up Profile of Ground States of Boson Stars with External Potential, J. Stat. Phys., 169 (2017), 395-422.  doi: 10.1007/s10955-017-1872-1.

[22]

F. Pusateri, Modified Scattering for the Boson Star Equation, Commun. Math. Phys., 332 (2014), 1203-1234.  doi: 10.1007/s00220-014-2094-x.

[23]

Q. Shi and C. Peng, Well-posedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonl. Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.

[24]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differ. Equ., 269 (2020), 6941-6987.  doi: 10.1016/j.jde.2020.05.016.

[25]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 1-43.  doi: 10.1016/j.jfa.2020.108610.

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[27]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Partial Differ. Equ., 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.

[28]

C. Topaz, A. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[29]

Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differ. Equ., 262 (2017), 2684-2704.  doi: 10.1016/j.jde.2016.11.004.

[30]

J. Yang and J. Yang, Existence and mass concentration of pseudo-relativistic Hartree equation, J. Math. Phys., 58 (2017), 1-22.  doi: 10.1063/1.4996576.

[31]

V. C. Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Ibero., 29 (2013), 1421-1436.  doi: 10.4171/RMI/763.

[32]

X. Zeng and L. Zhang, Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials, J. Math. Anal. Appl., 452 (2017), 47-61.  doi: 10.1016/j.jmaa.2017.02.053.

[1]

Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71
[2]

Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
[3]

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

Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[5]

Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261
[6]

Xiaomei Chen, Xiaohui Yu. Liouville type theorem for Hartree-Fock Equation on half space. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2079-2100. doi: 10.3934/cpaa.2022050
[7]

Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
[8]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[9]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
[10]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[11]

Tarek Saanouni. Energy scattering for the focusing fractional generalized Hartree equation. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3637-3654. doi: 10.3934/cpaa.2021124
[12]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations and Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531
[13]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
[14]

Kenji Nakanishi. Modified wave operators for the Hartree equation with data, image and convergence in the same space. Communications on Pure and Applied Analysis, 2002, 1 (2) : 237-252. doi: 10.3934/cpaa.2002.1.237
[15]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037
[16]

Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209
[17]

Yonggeun Cho, Tohru Ozawa, Hironobu Sasaki, Yongsun Shim. Remarks on the semirelativistic Hartree equations. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1277-1294. doi: 10.3934/dcds.2009.23.1277
[18]

Antonio Iannizzotto, Kanishka Perera, Marco Squassina. Ground states for scalar field equations with anisotropic nonlocal nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5963-5976. doi: 10.3934/dcds.2015.35.5963
[19]

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

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (206)
  • HTML views (150)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy