# American Institute of Mathematical Sciences

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

 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 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)} $. Citation: Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted$ 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. 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