Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity

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January 2014, 13(1): 273-291. doi: 10.3934/cpaa.2014.13.273

Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity

Tarek Saanouni ^1,

1.	University Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, 2092, Tunis, Tunisia

Received December 2012 Revised May 2013 Published July 2013

Abstract
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Extending previous works [47, 27, 30], we consider in even space dimensions the initial value problems for some high-order semi-linear wave and Schrödinger type equations with exponential nonlinearity. We obtain global well-posedness in the energy space.

Keywords: high-order Schrödinger type equation, High-order wave type equation, Moser-Trudinger inequality, well-posedness., energy estimate.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B3.

Citation: Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

References:

[1]	S. Adachi and K. Tanaka, Trudinger type inequalities in $R^N$ and their best exponent, Proc. Amer. Math. Society, 128 (1999), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.
[2]	A. Atallah Baraket, Local existence and estimations for a semilinear wave equation in two dimension space, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat, 8 (2004), 1-21. doi: MR204459.
[3]	J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0.
[4]	T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Metodos Matematicos, 26 (1996), Instituto de Matematica UFRJ.
[5]	T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonl. Anal. - TMA, 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.
[6]	J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions, J. Hyperbolic Differ. Equ, 6 (2009), 549-575. doi: 10.1142/S0219891609001927.
[7]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $R^3$, Ann. Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.
[8]	G. M. Constantine and T. H. Savitis, A multivariate Faa Di Bruno formula with applications, T. A. M. S, 348 (1996), 503-520.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates, A. M. S, 120 (1998), 955-980. doi: 10.1016/0362-546X(90)90023-A.
[10]	V. A. Galaktionov and S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Analysis, 53 (2003), 453-466. doi: 10.1016/S0362-546X(02)00311-5.
[11]	J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z, 189 (1985), 487-505. doi: 10.1007/BF01168155.
[12]	M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Annal. of Math., 132 (1990), 485-509. doi: 10.2307/1971427.
[13]	E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations, (2008).
[14]	S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear $2D$ Klein-Gordon equation with exponential type nonlinearity, Comm. Pure App. Math, 59 (2006), 1639-1658. doi: 10.1002/cpa.20127.
[15]	S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves, C. R. Acad. Sci. Paris, ser. I, 345 (2007), 133-138. doi: 10.1016/j.crma.2007.06.008.
[16]	S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the $2D$ energy critical wave equation, Duke Math., 150 (2009), 287-329. doi: 10.1215/00127094-2009-053.
[17]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion fourth-order nonlinear Schrödinger equations, Phys. Rev. E, 53 (1996), 1336-1339. doi: 10.1103/PhysRevE.53.R1336.
[18]	V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.
[19]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s11511-008-0031-6.
[20]	J. Kim, A. Arnold and X. Yao, Global estimates of fundamental solutions for higher-order Schrödinger equations,, \arXiv{0807.0690v2}., ().
[21]	J. Kim, A. Arnold and X. Yao, Estimates for a class of oscillatory integrals and decay rates for wave-type equations,, \arXiv{1109.0452v2}, ().
[22]	S. P. Levandosky, Stability and instability of fourth-order solitary waves, J. Dynam. Differential Equations, 10 (1998), 151-188. doi: 1040-7294/98/0100-0151S15.00/0.
[23]	S. P. Levandosky, Deacy estimates for fourth-order wave equations, J. Differential Equations, 143 (1998), 360-413. doi: 10.1006/jdeq.1997.3369.
[24]	S. P. Levandosky and W. A. Strauss, Time decay for the nonlinear beam equation, Methods and Applications of Analysis, 7 (2000), 479-488. doi: Zbl 1212.35476.
[25]	H. A Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t} = ..Au + F(u)$, T. A. M. S, 192 (1974), 1-21.
[26]	G. Lebeau, Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège, 70 (2001), 267-306.
[27]	G. Lebeau, Perte de régularité pour l'équation des ondes surcritique, Bull. Soc. Math. France, 133 (2005), 145-157.
[28]	J. L. Lions, Une remarque sur les problèmes d'évolution non linéaires dans des domaines non cylindriques, Revue Roumaine Math. Pur. Appl., 9 (1964), 129-135.
[29]	O. Mahouachi and T. Saanouni, Global well posedness and linearization of a semilinear wave equation with exponential growth, Georgian Math. J., 17 (2010), 543-562. doi: 10.1515/gmj.2010.026.
[30]	O. Mahouachi and T. Saanouni, Well and ill posedness issues for a class of $2D$ wave equation with exponential nonlinearity, J. P. D. E, 24 (2011), 361-384. doi: 10.4208/jpde.v24.n4.7.
[31]	M. Majdoub and T. Saanouni, Global well-posedness of some critical fourth-order wave and Schrödinger equation,, preprint., ().
[32]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth-order in the radial case,, \arXiv{0807.0690v2 }., ().
[33]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth-order in dimensions $d\geq9$,, \arXiv{0807.0692v2}., ().
[34]	J. Moser, A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J., 20 (1971), 1077-1092.
[35]	M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, Journal of Functional Analysis, 155 (1998), 364-380. doi: 10.1006/jfan.1997.3236.
[36]	M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z, 231 (1999), 479-487. doi: 10.1007/PL00004737.
[37]	T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl., 155 (1991), 531-540.
[38]	T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.
[39]	B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 01/2007.4:197-225.
[40]	B. Pausader, The cubic fourth-order Schrödinger equation, Journal of Functional Analysis, 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.
[41]	B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations, J. Differential Equations, 241 (2007), 237-278. doi: 10.1016/j.jde.2007.06.001.
[42]	B. Pausader and W. Strauss, Analyticity of the scattering operator for the beam equation, Discrete Contin. Dyn. Syst., 25 (2009), 617-626. doi: 10.3934/dcds.2009.25.617.
[43]	L. Peletier and W. C. Troy, Higher order models in Physics and Mechanics, Prog in Non Diff Eq and App, 45 (2001).
[44]	B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $R^2$, J. Funct. Analysis, 219 (2004), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[45]	B. Ruf and S. Sani, Sharp Adams-type inequalities in $R^n$, Trans. Amer. Math. Soc., 365 (2013), 645-670. doi: 10.1090/S0002-9947-2012-05561-9.
[46]	E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $R^{1+4}$, Amer. J. Math., 129 (2007), 1-60.
[47]	T. Saanouni, Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth, Bull. Belg. Math. Soc., 17 (2010), 441-462.
[48]	T. Saanouni, Decay of Solutions to a $2D$ Schrödinger Equation, J. Part. Diff. Eq., 24 (2011), 37-54. doi: 10.4208/jpde.v24.n1.3.
[49]	T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space, Math. Meth. Appl. Sci, 33 (2010), 1046-1058. doi: 10.1002/mma.1237.
[50]	T. Saanouni, Remarks on the semilinear Schrödinger equation, J. Math. Anal. Appl., 400 (2013), 331-344. doi: 10.1016/j.jmaa.2012.11.037.
[51]	I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129-135.
[52]	Shangbin Cui and Cuihua Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(\mathbbR^n)$ and applications, Nonlinear Analysis, 67 (2007), 687-707. doi: 10.1016/j.na.2006.06.020.
[53]	J. Shatah et M. Struwe, Well-posedness in the energy space for semilinear wave equation with critical growth, IMRN, 7 (1994), 303-309. doi: 10.1155/S1073792894000346.
[54]	W. A. Strauss, On weak solutions of semi-linear hyperbolic equations, Anais Acad. Brasil. Cienc., 42 (1970), 645-651.
[55]	M. Struwe, Semilinear wave equations, Bull. Amer. Math. Soc, N.S, 26 (1992), 53-85.
[56]	M. Struwe, The critical nonlinear wave equation in 2 space dimensions,, J. European Math. Soc. (to appear)., ().
[57]	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.
[58]	T. Tao, Global well-posedness and scattering for the higher-dimensional energycritical non-linear Schrödinger equation for radial data, New York J. of Math., 11 (2005), 57-80.
[59]	N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.
[60]	M. Visan, The defocusing energy-critical nolinear Schrödinger equation in higher dimensions, Duke. Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0.

show all references

References:

[1]	S. Adachi and K. Tanaka, Trudinger type inequalities in $R^N$ and their best exponent, Proc. Amer. Math. Society, 128 (1999), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.
[2]	A. Atallah Baraket, Local existence and estimations for a semilinear wave equation in two dimension space, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat, 8 (2004), 1-21. doi: MR204459.
[3]	J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0.
[4]	T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Metodos Matematicos, 26 (1996), Instituto de Matematica UFRJ.
[5]	T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonl. Anal. - TMA, 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.
[6]	J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions, J. Hyperbolic Differ. Equ, 6 (2009), 549-575. doi: 10.1142/S0219891609001927.
[7]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $R^3$, Ann. Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.
[8]	G. M. Constantine and T. H. Savitis, A multivariate Faa Di Bruno formula with applications, T. A. M. S, 348 (1996), 503-520.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates, A. M. S, 120 (1998), 955-980. doi: 10.1016/0362-546X(90)90023-A.
[10]	V. A. Galaktionov and S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Analysis, 53 (2003), 453-466. doi: 10.1016/S0362-546X(02)00311-5.
[11]	J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z, 189 (1985), 487-505. doi: 10.1007/BF01168155.
[12]	M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Annal. of Math., 132 (1990), 485-509. doi: 10.2307/1971427.
[13]	E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations, (2008).
[14]	S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear $2D$ Klein-Gordon equation with exponential type nonlinearity, Comm. Pure App. Math, 59 (2006), 1639-1658. doi: 10.1002/cpa.20127.
[15]	S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves, C. R. Acad. Sci. Paris, ser. I, 345 (2007), 133-138. doi: 10.1016/j.crma.2007.06.008.
[16]	S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the $2D$ energy critical wave equation, Duke Math., 150 (2009), 287-329. doi: 10.1215/00127094-2009-053.
[17]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion fourth-order nonlinear Schrödinger equations, Phys. Rev. E, 53 (1996), 1336-1339. doi: 10.1103/PhysRevE.53.R1336.
[18]	V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.
[19]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s11511-008-0031-6.
[20]	J. Kim, A. Arnold and X. Yao, Global estimates of fundamental solutions for higher-order Schrödinger equations,, \arXiv{0807.0690v2}., ().
[21]	J. Kim, A. Arnold and X. Yao, Estimates for a class of oscillatory integrals and decay rates for wave-type equations,, \arXiv{1109.0452v2}, ().
[22]	S. P. Levandosky, Stability and instability of fourth-order solitary waves, J. Dynam. Differential Equations, 10 (1998), 151-188. doi: 1040-7294/98/0100-0151S15.00/0.
[23]	S. P. Levandosky, Deacy estimates for fourth-order wave equations, J. Differential Equations, 143 (1998), 360-413. doi: 10.1006/jdeq.1997.3369.
[24]	S. P. Levandosky and W. A. Strauss, Time decay for the nonlinear beam equation, Methods and Applications of Analysis, 7 (2000), 479-488. doi: Zbl 1212.35476.
[25]	H. A Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t} = ..Au + F(u)$, T. A. M. S, 192 (1974), 1-21.
[26]	G. Lebeau, Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège, 70 (2001), 267-306.
[27]	G. Lebeau, Perte de régularité pour l'équation des ondes surcritique, Bull. Soc. Math. France, 133 (2005), 145-157.
[28]	J. L. Lions, Une remarque sur les problèmes d'évolution non linéaires dans des domaines non cylindriques, Revue Roumaine Math. Pur. Appl., 9 (1964), 129-135.
[29]	O. Mahouachi and T. Saanouni, Global well posedness and linearization of a semilinear wave equation with exponential growth, Georgian Math. J., 17 (2010), 543-562. doi: 10.1515/gmj.2010.026.
[30]	O. Mahouachi and T. Saanouni, Well and ill posedness issues for a class of $2D$ wave equation with exponential nonlinearity, J. P. D. E, 24 (2011), 361-384. doi: 10.4208/jpde.v24.n4.7.
[31]	M. Majdoub and T. Saanouni, Global well-posedness of some critical fourth-order wave and Schrödinger equation,, preprint., ().
[32]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth-order in the radial case,, \arXiv{0807.0690v2 }., ().
[33]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth-order in dimensions $d\geq9$,, \arXiv{0807.0692v2}., ().
[34]	J. Moser, A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J., 20 (1971), 1077-1092.
[35]	M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, Journal of Functional Analysis, 155 (1998), 364-380. doi: 10.1006/jfan.1997.3236.
[36]	M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z, 231 (1999), 479-487. doi: 10.1007/PL00004737.
[37]	T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl., 155 (1991), 531-540.
[38]	T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.
[39]	B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 01/2007.4:197-225.
[40]	B. Pausader, The cubic fourth-order Schrödinger equation, Journal of Functional Analysis, 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.
[41]	B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations, J. Differential Equations, 241 (2007), 237-278. doi: 10.1016/j.jde.2007.06.001.
[42]	B. Pausader and W. Strauss, Analyticity of the scattering operator for the beam equation, Discrete Contin. Dyn. Syst., 25 (2009), 617-626. doi: 10.3934/dcds.2009.25.617.
[43]	L. Peletier and W. C. Troy, Higher order models in Physics and Mechanics, Prog in Non Diff Eq and App, 45 (2001).
[44]	B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $R^2$, J. Funct. Analysis, 219 (2004), 340-367. doi: 10.1016/j.jfa.2004.06.013.
[45]	B. Ruf and S. Sani, Sharp Adams-type inequalities in $R^n$, Trans. Amer. Math. Soc., 365 (2013), 645-670. doi: 10.1090/S0002-9947-2012-05561-9.
[46]	E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $R^{1+4}$, Amer. J. Math., 129 (2007), 1-60.
[47]	T. Saanouni, Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth, Bull. Belg. Math. Soc., 17 (2010), 441-462.
[48]	T. Saanouni, Decay of Solutions to a $2D$ Schrödinger Equation, J. Part. Diff. Eq., 24 (2011), 37-54. doi: 10.4208/jpde.v24.n1.3.
[49]	T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space, Math. Meth. Appl. Sci, 33 (2010), 1046-1058. doi: 10.1002/mma.1237.
[50]	T. Saanouni, Remarks on the semilinear Schrödinger equation, J. Math. Anal. Appl., 400 (2013), 331-344. doi: 10.1016/j.jmaa.2012.11.037.
[51]	I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129-135.
[52]	Shangbin Cui and Cuihua Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(\mathbbR^n)$ and applications, Nonlinear Analysis, 67 (2007), 687-707. doi: 10.1016/j.na.2006.06.020.
[53]	J. Shatah et M. Struwe, Well-posedness in the energy space for semilinear wave equation with critical growth, IMRN, 7 (1994), 303-309. doi: 10.1155/S1073792894000346.
[54]	W. A. Strauss, On weak solutions of semi-linear hyperbolic equations, Anais Acad. Brasil. Cienc., 42 (1970), 645-651.
[55]	M. Struwe, Semilinear wave equations, Bull. Amer. Math. Soc, N.S, 26 (1992), 53-85.
[56]	M. Struwe, The critical nonlinear wave equation in 2 space dimensions,, J. European Math. Soc. (to appear)., ().
[57]	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.
[58]	T. Tao, Global well-posedness and scattering for the higher-dimensional energycritical non-linear Schrödinger equation for radial data, New York J. of Math., 11 (2005), 57-80.
[59]	N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.
[60]	M. Visan, The defocusing energy-critical nolinear Schrödinger equation in higher dimensions, Duke. Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0.

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