-
Previous Article
Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume
- CPAA Home
- This Issue
-
Next Article
Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc
Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity
1. | Laboratoire Paul Painlevé UMR 8524, Université Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France |
2. | Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France, Department of Mathematics, HCMC University of Pedagogy |
3. | Department of Mathematics, College of science, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, Saudi Arabia |
4. | Basic & Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia |
$ i \partial_tu + \Delta u = |x|^{-b} \left({\rm e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0) = u_0, \quad x \in \mathbb{R}^2, $ |
$ 0<b<1 $ |
$ \alpha = 2\pi(2-b) $ |
$ u_0 $ |
$ \Sigma(\mathbb{R}^2) = \{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\} $ |
$ \Sigma $ |
$ \frac{2}{(1+b)(2-b)} $ |
References:
[1] |
S. Adachi and K. Tanaka,
Trudinger type inequalities in $\mathbb R^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.
doi: 10.1090/S0002-9939-99-05180-1. |
[2] |
A. Adam Azzam, Doubly Critical Semilinear Schrödinger Equations, Ph.D dissertation, UCLA, 2017. Google Scholar |
[3] |
H. Bahouri, S. Ibrahim and G. Perelman,
Scattering for the critical 2-D NLS with exponential growth, Differential Integral Equations, 27 (2014), 233-268.
|
[4] |
C. Bennet and R. Sharply, Interpolation of Operators, Academic Press, Pure and Applied Mathematics 129, 1988. Google Scholar |
[5] |
A. Bensouilah, D. Draouil and M. Majdoub,
Energy critical Schrödinger equation with weighted exponential nonlinearity: Local and global well-posedness, J. Hyperbolic Differ. Equ., 15 (2018), 599-621.
doi: 10.1142/S0219891618500194. |
[6] |
J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
doi: 10.1007/978-3-642-66451-9. |
[7] |
T. Cazenave,
Equations de Schrödinger non linéaires en dimension deux, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 327-346.
doi: 10.1017/S0308210500017182. |
[8] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003. |
[9] |
J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi,
Energy critical NLS in two space dimension, J. Hyperbolic Differ. Equ., 6 (2009), 549-575.
doi: 10.1142/S0219891609001927. |
[10] |
V. D. Dinh, Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1710.01392. Google Scholar |
[11] |
V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., (2019), 1–24.
doi: 10.1007/s00028-019-00481-0. |
[12] |
E. Gagliardo,
Proprieta di alcune classi di funzioni in piu variabili, Ric. Mat., 7 (1958), 102-137.
|
[13] |
L. Grafakos, Classical Fourier Analysis, 2nd ed., Graduate texts in Mathematics, Vol. 249, Springer, New York, 2008. |
[14] |
R. Hunt, On $L^{p, q}$ spaces, L'Enseign. Math., 12 (1967), 249-276. Google Scholar |
[15] |
S. Ibrahim, M. Majdoub and N. Masmoudi,
Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc., 135 (2007), 87-97.
doi: 10.1090/S0002-9939-06-08240-2. |
[16] |
S. Ibrahim, M. Majdoub and N. Masmoudi,
Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math., 59 (2006), 1639-1658.
doi: 10.1002/cpa.20127. |
[17] |
S. Ibrahim, M. Majdoub and N. Masmoudi,
Well- and ill-posednessissues for energy supercritical waves, Analysis & PDE., 4 (2011), 341-367.
doi: 10.2140/apde.2011.4.341. |
[18] |
S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi,
Scattering for the two-dimensional energy-critical wave equation, Duke Math. J., 150 (2009), 287-329.
doi: 10.1215/00127094-2009-053. |
[19] |
S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi,
Scattering for the two-dimensional NLS with exponential nonlinearity, Nonlinearity, 25 (2012), 1843-1849.
doi: 10.1088/0951-7715/25/6/1843. |
[20] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[21] |
J. F. Lam, B. Lippman and an F. Tappert,
Self-trapped laser beams in plasma, Phys. Fluid., 20 (1977), 1176-1179.
doi: 10.1063/1.861679. |
[22] |
M. Nakamura and T. Ozawa,
Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal., 155 (1998), 364-380.
doi: 10.1006/jfan.1997.3236. |
[23] |
L. Nirenberg,
On elliptic partial differential equations (lecture Ⅱ), Ann. Sc. Norm. Super. Pisa, Cl. Sci., 13 (1959), 115-162.
|
[24] |
R. O'Neil,
Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.
doi: 10.1215/S0012-7094-63-03015-1. |
[25] |
F. Planchon and L. Vega,
Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 4 (2009), 261-290.
doi: 10.24033/asens.2096. |
[26] |
B. Ruf,
A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^{2}$, J. Funct. Anal., 219 (2005), 340-367.
doi: 10.1016/j.jfa.2004.06.013. |
[27] |
M. de Souza,
On a class of singular Trudinger-Moser type inequalities for unbounded domains in $\mathbb{R}^N$, Appl. Math. Lett., 25 (2012), 2100-2104.
doi: 10.1016/j.aml.2012.05.007. |
[28] |
M. de Souza and J. M. do Ò,
On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents, Potential Anal., 38 (2013), 1091-1101.
doi: 10.1007/s11118-012-9308-7. |
[29] | E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces, Princeton Mathematical Series, Princeton University Press, 1971. Google Scholar |
[30] |
M. Sack and M. Struwe,
Scattering for a critical nonlinear wave equation in two space dimensions, Math. Ann., 365 (2016), 969-985.
doi: 10.1007/s00208-015-1282-0. |
[31] |
M. Struwe,
Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann., 350 (2011), 707-719.
doi: 10.1007/s00208-010-0567-6. |
[32] |
M. Struwe,
The critical nonlinear wave equation in two space dimensions, J. Eur. Math. Soc., 15 (2013), 1805-1823.
doi: 10.4171/JEMS/404. |
[33] |
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: 10.1080/03605300701588805. |
[34] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS, Regional Conference Series in Mathematics, Number 106, American Mathematical Society, 2006.
doi: 10.1090/cbms/106. |
show all references
References:
[1] |
S. Adachi and K. Tanaka,
Trudinger type inequalities in $\mathbb R^N$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057.
doi: 10.1090/S0002-9939-99-05180-1. |
[2] |
A. Adam Azzam, Doubly Critical Semilinear Schrödinger Equations, Ph.D dissertation, UCLA, 2017. Google Scholar |
[3] |
H. Bahouri, S. Ibrahim and G. Perelman,
Scattering for the critical 2-D NLS with exponential growth, Differential Integral Equations, 27 (2014), 233-268.
|
[4] |
C. Bennet and R. Sharply, Interpolation of Operators, Academic Press, Pure and Applied Mathematics 129, 1988. Google Scholar |
[5] |
A. Bensouilah, D. Draouil and M. Majdoub,
Energy critical Schrödinger equation with weighted exponential nonlinearity: Local and global well-posedness, J. Hyperbolic Differ. Equ., 15 (2018), 599-621.
doi: 10.1142/S0219891618500194. |
[6] |
J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
doi: 10.1007/978-3-642-66451-9. |
[7] |
T. Cazenave,
Equations de Schrödinger non linéaires en dimension deux, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 327-346.
doi: 10.1017/S0308210500017182. |
[8] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003. |
[9] |
J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi,
Energy critical NLS in two space dimension, J. Hyperbolic Differ. Equ., 6 (2009), 549-575.
doi: 10.1142/S0219891609001927. |
[10] |
V. D. Dinh, Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1710.01392. Google Scholar |
[11] |
V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., (2019), 1–24.
doi: 10.1007/s00028-019-00481-0. |
[12] |
E. Gagliardo,
Proprieta di alcune classi di funzioni in piu variabili, Ric. Mat., 7 (1958), 102-137.
|
[13] |
L. Grafakos, Classical Fourier Analysis, 2nd ed., Graduate texts in Mathematics, Vol. 249, Springer, New York, 2008. |
[14] |
R. Hunt, On $L^{p, q}$ spaces, L'Enseign. Math., 12 (1967), 249-276. Google Scholar |
[15] |
S. Ibrahim, M. Majdoub and N. Masmoudi,
Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc., 135 (2007), 87-97.
doi: 10.1090/S0002-9939-06-08240-2. |
[16] |
S. Ibrahim, M. Majdoub and N. Masmoudi,
Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math., 59 (2006), 1639-1658.
doi: 10.1002/cpa.20127. |
[17] |
S. Ibrahim, M. Majdoub and N. Masmoudi,
Well- and ill-posednessissues for energy supercritical waves, Analysis & PDE., 4 (2011), 341-367.
doi: 10.2140/apde.2011.4.341. |
[18] |
S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi,
Scattering for the two-dimensional energy-critical wave equation, Duke Math. J., 150 (2009), 287-329.
doi: 10.1215/00127094-2009-053. |
[19] |
S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi,
Scattering for the two-dimensional NLS with exponential nonlinearity, Nonlinearity, 25 (2012), 1843-1849.
doi: 10.1088/0951-7715/25/6/1843. |
[20] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[21] |
J. F. Lam, B. Lippman and an F. Tappert,
Self-trapped laser beams in plasma, Phys. Fluid., 20 (1977), 1176-1179.
doi: 10.1063/1.861679. |
[22] |
M. Nakamura and T. Ozawa,
Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal., 155 (1998), 364-380.
doi: 10.1006/jfan.1997.3236. |
[23] |
L. Nirenberg,
On elliptic partial differential equations (lecture Ⅱ), Ann. Sc. Norm. Super. Pisa, Cl. Sci., 13 (1959), 115-162.
|
[24] |
R. O'Neil,
Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.
doi: 10.1215/S0012-7094-63-03015-1. |
[25] |
F. Planchon and L. Vega,
Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 4 (2009), 261-290.
doi: 10.24033/asens.2096. |
[26] |
B. Ruf,
A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb R}^{2}$, J. Funct. Anal., 219 (2005), 340-367.
doi: 10.1016/j.jfa.2004.06.013. |
[27] |
M. de Souza,
On a class of singular Trudinger-Moser type inequalities for unbounded domains in $\mathbb{R}^N$, Appl. Math. Lett., 25 (2012), 2100-2104.
doi: 10.1016/j.aml.2012.05.007. |
[28] |
M. de Souza and J. M. do Ò,
On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents, Potential Anal., 38 (2013), 1091-1101.
doi: 10.1007/s11118-012-9308-7. |
[29] | E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces, Princeton Mathematical Series, Princeton University Press, 1971. Google Scholar |
[30] |
M. Sack and M. Struwe,
Scattering for a critical nonlinear wave equation in two space dimensions, Math. Ann., 365 (2016), 969-985.
doi: 10.1007/s00208-015-1282-0. |
[31] |
M. Struwe,
Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann., 350 (2011), 707-719.
doi: 10.1007/s00208-010-0567-6. |
[32] |
M. Struwe,
The critical nonlinear wave equation in two space dimensions, J. Eur. Math. Soc., 15 (2013), 1805-1823.
doi: 10.4171/JEMS/404. |
[33] |
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: 10.1080/03605300701588805. |
[34] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS, Regional Conference Series in Mathematics, Number 106, American Mathematical Society, 2006.
doi: 10.1090/cbms/106. |
[1] |
Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110 |
[2] |
Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212 |
[3] |
Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011 |
[4] |
Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963 |
[5] |
Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064 |
[6] |
Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121 |
[7] |
Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 |
[8] |
Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378 |
[9] |
Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 |
[10] |
Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193 |
[11] |
Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006 |
[12] |
Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803 |
[13] |
Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 |
[14] |
Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031 |
[15] |
Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323 |
[16] |
Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016 |
[17] |
Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 |
[18] |
Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481 |
[19] |
Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091 |
[20] |
J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445 |
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]