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Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case
Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system
School of Artificial Intelligence and Information Technology, Nanjing University of Chinese Medicine, Nanjing 210046, China |
$\begin{eqnarray}i\partial_{t}\phi+\Delta\phi = \frac{2m}{r^2}A_{\theta}\phi+A_{0}\phi+\frac{1}{r^2}A_{\theta}^2\phi-\lambda|\phi|^{p-2}\phi,\\ \partial_rA_{0} = \frac{1}{r}(m+A_{\theta})|\phi|^2,\\ \partial_tA_{\theta} = rIm(\bar{\phi}\partial_{r}\phi),\\ \partial_rA_{\theta} = -\frac{1}{2}|\phi|^2r,\\ A_r = 0.\end{eqnarray}$ |
$ \phi(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R} $ |
$ A_\mu(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R} $ |
$ \mu = 0, 1, 2 $ |
$ A_r = \frac{x_1}{|x|}A_1+\frac{x_2}{|x|}A_2 $ |
$ A_{\theta} = -x_2A_1+x_1A_2 $ |
$ \lambda<0 $ |
$ p>4 $ |
$ p>4 $ |
References:
[1] |
H. Bahouri and P. Gérard,
High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[2] |
L. Bergé, A. De Bouard and J.-C. Saut,
Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.
doi: 10.1088/0951-7715/8/2/007. |
[3] |
J. Byeon, H. Huh and J. Seok,
Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.
doi: 10.1016/j.jfa.2012.05.024. |
[4] |
J. Byeon, H. Huh and J. Seok,
On standing waves with a vortex point of order $N$ for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.
doi: 10.1016/j.jde.2016.04.004. |
[5] |
X. Cheng, C. Miao and L. Zhao,
Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934.
doi: 10.1016/j.jde.2016.04.031. |
[6] |
G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Physics Monographs, 36, Springer, Berlin, Heidelberg, 1995.
doi: 10.1007/978-3-540-44777-1. |
[7] |
T. Duyckaerts, J. Holmer and S. Roudenko,
Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[8] |
D. Fang, J. Xie and T. Cazenave,
Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[9] |
J. Ginibre and G. Velo,
Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pur. Appl. (9), 64 (1985), 363-401.
|
[10] |
H. Huh,
Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.
doi: 10.1088/0951-7715/22/5/003. |
[11] |
H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53 (2012), 8pp.
doi: 10.1063/1.4726192. |
[12] |
H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), 7pp.
doi: 10.1155/2013/590653. |
[13] |
J. Holmer and S. Roudenko,
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[14] |
R. Jackiw and S.-Y. Pi,
Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D (3), 42 (1990), 3500-3513.
doi: 10.1103/PhysRevD.42.3500. |
[15] |
R. Jackiw and S.-Y. Pi,
Self-dual Chern-Simons solitons, Prog. Theor. Phys. Suppl., 107 (1992), 1-40.
doi: 10.1143/PTPS.107.1. |
[16] |
Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrödinger equation with a vortex point, Commun. Contemp. Math., 18 (2016), 20pp.
doi: 10.1142/S0219199715500741. |
[17] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[18] |
G. Li and X. Luo,
Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^2$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428.
doi: 10.5186/aasfm.2017.4223. |
[19] |
Z. M. Lim,
Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system, J. Differential Equations, 264 (2018), 2553-2597.
doi: 10.1016/j.jde.2017.10.026. |
[20] |
B. Liu and P. Smith,
Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.
doi: 10.4171/RMI/898. |
[21] |
B. Liu, P. Smith and D. Tataru,
Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not. IMRN, 2014 (2013), 6341-6398.
doi: 10.1093/imrn/rnt161. |
[22] |
F. Merle and L. Vega,
Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, 1998 (1998), 399-425.
doi: 10.1155/S1073792898000270. |
[23] |
K. Nakanishi,
Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225.
doi: 10.1006/jfan.1999.3503. |
[24] |
S.-J. Oh and F. Pusateri,
Decay and scattering for the Chern-Simons-Schrödinger equations, Int. Math. Res. Not. IMRN, 2015 (2015), 13122-13147.
doi: 10.1093/imrn/rnv093. |
[25] |
A. Pomponio and D. Ruiz,
A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 17 (2015), 1463-1486.
doi: 10.4171/JEMS/535. |
[26] |
K. Sahbi,
On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.
doi: 10.1006/jdeq.2000.3951. |
[27] |
J. Yuan,
Multiple normalized solutions of Chern-Simons-Schrödinger system, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1801-1816.
doi: 10.1007/s00030-015-0344-z. |
[28] |
J. Yuan, Global existence and scattering of radial Chern-Simons-Schrödinger system, submitted. |
show all references
References:
[1] |
H. Bahouri and P. Gérard,
High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[2] |
L. Bergé, A. De Bouard and J.-C. Saut,
Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.
doi: 10.1088/0951-7715/8/2/007. |
[3] |
J. Byeon, H. Huh and J. Seok,
Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.
doi: 10.1016/j.jfa.2012.05.024. |
[4] |
J. Byeon, H. Huh and J. Seok,
On standing waves with a vortex point of order $N$ for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.
doi: 10.1016/j.jde.2016.04.004. |
[5] |
X. Cheng, C. Miao and L. Zhao,
Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934.
doi: 10.1016/j.jde.2016.04.031. |
[6] |
G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Physics Monographs, 36, Springer, Berlin, Heidelberg, 1995.
doi: 10.1007/978-3-540-44777-1. |
[7] |
T. Duyckaerts, J. Holmer and S. Roudenko,
Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[8] |
D. Fang, J. Xie and T. Cazenave,
Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[9] |
J. Ginibre and G. Velo,
Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pur. Appl. (9), 64 (1985), 363-401.
|
[10] |
H. Huh,
Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.
doi: 10.1088/0951-7715/22/5/003. |
[11] |
H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53 (2012), 8pp.
doi: 10.1063/1.4726192. |
[12] |
H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), 7pp.
doi: 10.1155/2013/590653. |
[13] |
J. Holmer and S. Roudenko,
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[14] |
R. Jackiw and S.-Y. Pi,
Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D (3), 42 (1990), 3500-3513.
doi: 10.1103/PhysRevD.42.3500. |
[15] |
R. Jackiw and S.-Y. Pi,
Self-dual Chern-Simons solitons, Prog. Theor. Phys. Suppl., 107 (1992), 1-40.
doi: 10.1143/PTPS.107.1. |
[16] |
Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrödinger equation with a vortex point, Commun. Contemp. Math., 18 (2016), 20pp.
doi: 10.1142/S0219199715500741. |
[17] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[18] |
G. Li and X. Luo,
Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^2$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428.
doi: 10.5186/aasfm.2017.4223. |
[19] |
Z. M. Lim,
Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system, J. Differential Equations, 264 (2018), 2553-2597.
doi: 10.1016/j.jde.2017.10.026. |
[20] |
B. Liu and P. Smith,
Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.
doi: 10.4171/RMI/898. |
[21] |
B. Liu, P. Smith and D. Tataru,
Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not. IMRN, 2014 (2013), 6341-6398.
doi: 10.1093/imrn/rnt161. |
[22] |
F. Merle and L. Vega,
Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, 1998 (1998), 399-425.
doi: 10.1155/S1073792898000270. |
[23] |
K. Nakanishi,
Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225.
doi: 10.1006/jfan.1999.3503. |
[24] |
S.-J. Oh and F. Pusateri,
Decay and scattering for the Chern-Simons-Schrödinger equations, Int. Math. Res. Not. IMRN, 2015 (2015), 13122-13147.
doi: 10.1093/imrn/rnv093. |
[25] |
A. Pomponio and D. Ruiz,
A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 17 (2015), 1463-1486.
doi: 10.4171/JEMS/535. |
[26] |
K. Sahbi,
On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.
doi: 10.1006/jdeq.2000.3951. |
[27] |
J. Yuan,
Multiple normalized solutions of Chern-Simons-Schrödinger system, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1801-1816.
doi: 10.1007/s00030-015-0344-z. |
[28] |
J. Yuan, Global existence and scattering of radial Chern-Simons-Schrödinger system, submitted. |
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