-
Previous Article
Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy
- CPAA Home
- This Issue
-
Next Article
Positive radial solutions of a nonlinear boundary value problem
On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China |
$i\partial_t u-(-Δ)^su+λ_1|u|^{2p_1}u+λ_2|u|^{2p_2}u = 0, $ |
$ 0<p_1<p_2<\frac{2s}{N-2s}$ |
$ 0<p_1<\frac{2s}{N}$ |
$p_2 = \frac{2s}{N}$ |
$L^2$ |
References:
[1] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.
doi: 10.1016/j.jde.2003.12.002. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. |
[3] |
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa,
On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224.
doi: 10.1016/j.jde.2003.12.002. |
[4] |
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa,
On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.
doi: 10.1016/j.jde.2003.12.002. |
[5] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Profile decompositions and Blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29.
doi: 10.1016/j.jde.2003.12.002. |
[6] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479.
doi: 10.1016/j.jde.2003.12.002. |
[7] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.
doi: 10.1016/j.jde.2003.12.002. |
[8] |
B. Feng,
Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential, Nonlinear Anal. Real World Appl., 31 (2016), 132-145.
doi: 10.1016/j.jde.2003.12.002. |
[9] |
B. Feng,
On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, Journal of Evolution Equations, 18 (2018), 203-220.
doi: 10.1016/j.jde.2003.12.002. |
[10] |
B. Feng and Y. Cai,
Concentration for blow-up solutions of the Davey-Stewartson system in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 26 (2015), 330-342.
doi: 10.1016/j.jde.2003.12.002. |
[11] |
B. Feng and X. Yuan,
On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.
doi: 10.1016/j.jde.2003.12.002. |
[12] |
B. Feng and H. Zhang,
Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364.
doi: 10.1016/j.jde.2003.12.002. |
[13] |
B. Feng and H. Zhang,
Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507.
doi: 10.1016/j.jde.2003.12.002. |
[14] |
B. Feng, D. Zhao and C. Sun,
On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain, J. Math. Anal. Appl., 416 (2014), 901-923.
doi: 10.1016/j.jde.2003.12.002. |
[15] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
doi: 10.1016/j.jde.2003.12.002. |
[16] |
R. T. Glassey,
On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1016/j.jde.2003.12.002. |
[17] |
Q. Guo and S. Zhu,
Sharp threshold of blow-up and scattering for the fractional Hartree equation, Journal of Differential Equations, 264 (2018), 2802-2832.
doi: 10.1016/j.jde.2003.12.002. |
[18] |
T. Hmidi and S. Keraani,
Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828.
doi: 10.1016/j.jde.2003.12.002. |
[19] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.
doi: 10.1016/j.jde.2003.12.002. |
[20] |
C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364.
doi: 10.1016/j.jde.2003.12.002. |
[21] |
E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal and S. K. Turitsyn,
Sharper criteria for the wave collapse, Physica D, 87 (1995), 273-284.
doi: 10.1016/j.jde.2003.12.002. |
[22] |
N. Laskin,
Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304.
doi: 10.1016/j.jde.2003.12.002. |
[23] |
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108.
doi: 10.1016/j.jde.2003.12.002. |
[24] |
J. Liu and A. Qian,
Ground state solution for a Schrödinger-Poisson equation with critical growth, Nonlinear Anal. Real World Appl., 40 (2018), 428-443.
doi: 10.1016/j.jde.2003.12.002. |
[25] |
A. Mao, L. Yang, A. Qian and S. Luan,
Shixia Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett., 68 (2017), 8-12.
doi: 10.1016/j.jde.2003.12.002. |
[26] |
F. Merle and P. Raphaël,
On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-572.
doi: 10.1016/j.jde.2003.12.002. |
[27] |
F. Merle and P. Raphaël,
Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 16 (2005), 157-222.
doi: 10.1016/j.jde.2003.12.002. |
[28] |
F. Merle and P. Raphaël,
On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Amer. Soc., 19 (2006), 37-90.
doi: 10.1016/j.jde.2003.12.002. |
[29] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/j.jde.2003.12.002. |
[30] |
C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0873-0. |
[31] |
T. Tao, M. Visan and X. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1016/j.jde.2003.12.002. |
[32] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
doi: 10.1016/j.jde.2003.12.002. |
[33] |
M. I. Weinstein,
On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.
doi: 10.1016/j.jde.2003.12.002. |
[34] |
J. Zhang,
Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal., 48 (2002), 191-207.
doi: 10.1016/j.jde.2003.12.002. |
[35] |
J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030
doi: 10.1016/j.jde.2003.12.002. |
[36] |
J. Zhang and S. Zhu,
Sharp blow-up criteria for the Davey-Stewartson system in $\mathbb{R}^3$, Dynamics of PDE, 8 (2011), 239-260.
doi: 10.1016/j.jde.2003.12.002. |
[37] |
S. Zhu,
On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531.
doi: 10.1016/j.jde.2003.12.002. |
[38] |
S. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys., 57 (2016), 031501.
doi: 10.1016/j.jde.2003.12.002. |
[39] |
S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021
doi: 10.1016/j.jde.2003.12.002. |
show all references
References:
[1] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.
doi: 10.1016/j.jde.2003.12.002. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. |
[3] |
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa,
On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224.
doi: 10.1016/j.jde.2003.12.002. |
[4] |
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa,
On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.
doi: 10.1016/j.jde.2003.12.002. |
[5] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Profile decompositions and Blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29.
doi: 10.1016/j.jde.2003.12.002. |
[6] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479.
doi: 10.1016/j.jde.2003.12.002. |
[7] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.
doi: 10.1016/j.jde.2003.12.002. |
[8] |
B. Feng,
Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential, Nonlinear Anal. Real World Appl., 31 (2016), 132-145.
doi: 10.1016/j.jde.2003.12.002. |
[9] |
B. Feng,
On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, Journal of Evolution Equations, 18 (2018), 203-220.
doi: 10.1016/j.jde.2003.12.002. |
[10] |
B. Feng and Y. Cai,
Concentration for blow-up solutions of the Davey-Stewartson system in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 26 (2015), 330-342.
doi: 10.1016/j.jde.2003.12.002. |
[11] |
B. Feng and X. Yuan,
On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.
doi: 10.1016/j.jde.2003.12.002. |
[12] |
B. Feng and H. Zhang,
Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364.
doi: 10.1016/j.jde.2003.12.002. |
[13] |
B. Feng and H. Zhang,
Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507.
doi: 10.1016/j.jde.2003.12.002. |
[14] |
B. Feng, D. Zhao and C. Sun,
On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain, J. Math. Anal. Appl., 416 (2014), 901-923.
doi: 10.1016/j.jde.2003.12.002. |
[15] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
doi: 10.1016/j.jde.2003.12.002. |
[16] |
R. T. Glassey,
On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1016/j.jde.2003.12.002. |
[17] |
Q. Guo and S. Zhu,
Sharp threshold of blow-up and scattering for the fractional Hartree equation, Journal of Differential Equations, 264 (2018), 2802-2832.
doi: 10.1016/j.jde.2003.12.002. |
[18] |
T. Hmidi and S. Keraani,
Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828.
doi: 10.1016/j.jde.2003.12.002. |
[19] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.
doi: 10.1016/j.jde.2003.12.002. |
[20] |
C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364.
doi: 10.1016/j.jde.2003.12.002. |
[21] |
E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal and S. K. Turitsyn,
Sharper criteria for the wave collapse, Physica D, 87 (1995), 273-284.
doi: 10.1016/j.jde.2003.12.002. |
[22] |
N. Laskin,
Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304.
doi: 10.1016/j.jde.2003.12.002. |
[23] |
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108.
doi: 10.1016/j.jde.2003.12.002. |
[24] |
J. Liu and A. Qian,
Ground state solution for a Schrödinger-Poisson equation with critical growth, Nonlinear Anal. Real World Appl., 40 (2018), 428-443.
doi: 10.1016/j.jde.2003.12.002. |
[25] |
A. Mao, L. Yang, A. Qian and S. Luan,
Shixia Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett., 68 (2017), 8-12.
doi: 10.1016/j.jde.2003.12.002. |
[26] |
F. Merle and P. Raphaël,
On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-572.
doi: 10.1016/j.jde.2003.12.002. |
[27] |
F. Merle and P. Raphaël,
Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 16 (2005), 157-222.
doi: 10.1016/j.jde.2003.12.002. |
[28] |
F. Merle and P. Raphaël,
On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Amer. Soc., 19 (2006), 37-90.
doi: 10.1016/j.jde.2003.12.002. |
[29] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/j.jde.2003.12.002. |
[30] |
C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0873-0. |
[31] |
T. Tao, M. Visan and X. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1016/j.jde.2003.12.002. |
[32] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
doi: 10.1016/j.jde.2003.12.002. |
[33] |
M. I. Weinstein,
On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.
doi: 10.1016/j.jde.2003.12.002. |
[34] |
J. Zhang,
Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal., 48 (2002), 191-207.
doi: 10.1016/j.jde.2003.12.002. |
[35] |
J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030
doi: 10.1016/j.jde.2003.12.002. |
[36] |
J. Zhang and S. Zhu,
Sharp blow-up criteria for the Davey-Stewartson system in $\mathbb{R}^3$, Dynamics of PDE, 8 (2011), 239-260.
doi: 10.1016/j.jde.2003.12.002. |
[37] |
S. Zhu,
On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531.
doi: 10.1016/j.jde.2003.12.002. |
[38] |
S. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys., 57 (2016), 031501.
doi: 10.1016/j.jde.2003.12.002. |
[39] |
S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021
doi: 10.1016/j.jde.2003.12.002. |
[1] |
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 |
[2] |
Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 |
[3] |
Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 |
[4] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
[5] |
Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 |
[6] |
Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 |
[7] |
J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445 |
[8] |
Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 |
[9] |
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 |
[10] |
Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435 |
[11] |
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
[12] |
Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169 |
[13] |
Jinmyong An, Roesong Jang, Jinmyong Kim. Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022111 |
[14] |
Li Li. An inverse problem for a fractional diffusion equation with fractional power type nonlinearities. Inverse Problems and Imaging, 2022, 16 (3) : 613-624. doi: 10.3934/ipi.2021064 |
[15] |
Mingqi Xiang, Die Hu. Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125 |
[16] |
Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082 |
[17] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
[18] |
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 |
[19] |
Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure and Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435 |
[20] |
Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]