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September  2019, 18(5): 2735-2755. doi: 10.3934/cpaa.2019122

Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

Abdelwahab Bensouilah 1, , Van Duong Dinh 2,, and  Mohamed Majdoub 3,4,

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

* Corresponding author: Van Duong Dinh

Received  October 2018 Revised  January 2019 Published  April 2019

We investigate the defocusing inhomogeneous nonlinear Schrödinger equation
$ 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, $
with
$ 0<b<1 $
 and
$ \alpha = 2\pi(2-b) $
. First we show the decay of global solutions by assuming that the initial data
$ u_0 $
 belongs to the weighted space
$ \Sigma(\mathbb{R}^2) = \{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\} $
. Then we combine the local theory with the decay estimate to obtain scattering in
$ \Sigma $
 when the Hamiltonian is below the value
$ \frac{2}{(1+b)(2-b)} $
.
Keywords: Inhomogeneous nonlinear Schrödinger equation, virial identity, scattering, exponential nonlinearity, singular Moser-Trudinger inequality.
Mathematics Subject Classification: Primary: 35L70, 35Q55, 35B40; Secondary: 35B33, 37K05, 37L50.
Citation: Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6]

J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. doi: 10.1007/978-3-642-66451-9.  Google Scholar

[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.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. Gagliardo, Proprieta di alcune classi di funzioni in piu variabili, Ric. Mat., 7 (1958), 102-137.   Google Scholar

[13]

L. Grafakos, Classical Fourier Analysis, 2nd ed., Graduate texts in Mathematics, Vol. 249, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

L. Nirenberg, On elliptic partial differential equations (lecture Ⅱ), Ann. Sc. Norm. Super. Pisa, Cl. Sci., 13 (1959), 115-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6]

J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. doi: 10.1007/978-3-642-66451-9.  Google Scholar

[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.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. Gagliardo, Proprieta di alcune classi di funzioni in piu variabili, Ric. Mat., 7 (1958), 102-137.   Google Scholar

[13]

L. Grafakos, Classical Fourier Analysis, 2nd ed., Graduate texts in Mathematics, Vol. 249, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

L. Nirenberg, On elliptic partial differential equations (lecture Ⅱ), Ann. Sc. Norm. Super. Pisa, Cl. Sci., 13 (1959), 115-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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