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

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

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.
Citation: Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & 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. doi: 10.1090/S0002-9939-99-05180-1. Google Scholar

[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. doi: MR204459. Google Scholar

[3]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case,, J. Amer. Math. Soc., 12 (1999), 145. doi: 10.1090/S0894-0347-99-00283-0. Google Scholar

[4]

T. Cazenave, An introduction to nonlinear Schrödinger equations,, Textos de Metodos Matematicos, 26 (1996). Google Scholar

[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. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[6]

J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ, 6 (2009), 549. doi: 10.1142/S0219891609001927. Google Scholar

[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. doi: 10.4007/annals.2008.167.767. Google Scholar

[8]

G. M. Constantine and T. H. Savitis, A multivariate Faa Di Bruno formula with applications,, T. A. M. S, 348 (1996), 503. Google Scholar

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates,, A. M. S, 120 (1998), 955. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[10]

V. A. Galaktionov and S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations,, Nonlinear Analysis, 53 (2003), 453. doi: 10.1016/S0362-546X(02)00311-5. Google Scholar

[11]

J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation,, Math. Z, 189 (1985), 487. doi: 10.1007/BF01168155. Google Scholar

[12]

M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity,, Annal. of Math., 132 (1990), 485. doi: 10.2307/1971427. Google Scholar

[13]

E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations,, (2008)., (2008). Google Scholar

[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. doi: 10.1002/cpa.20127. Google Scholar

[15]

S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves,, C. R. Acad. Sci. Paris, 345 (2007), 133. doi: 10.1016/j.crma.2007.06.008. Google Scholar

[16]

S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the $2D$ energy critical wave equation,, Duke Math., 150 (2009), 287. doi: 10.1215/00127094-2009-053. Google Scholar

[17]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion fourth-order nonlinear Schrödinger equations,, Phys. Rev. E, 53 (1996), 1336. doi: 10.1103/PhysRevE.53.R1336. Google Scholar

[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. doi: 10.1016/S0167-2789(00)00078-6. Google Scholar

[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. doi: 10.1007/s11511-008-0031-6. Google Scholar

[20]

J. Kim, A. Arnold and X. Yao, Global estimates of fundamental solutions for higher-order Schrödinger equations,, \arXiv{0807.0690v2}., (). Google Scholar

[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}, (). Google Scholar

[22]

S. P. Levandosky, Stability and instability of fourth-order solitary waves,, J. Dynam. Differential Equations, 10 (1998), 151. doi: 1040-7294/98/0100-0151S15.00/0. Google Scholar

[23]

S. P. Levandosky, Deacy estimates for fourth-order wave equations,, J. Differential Equations, 143 (1998), 360. doi: 10.1006/jdeq.1997.3369. Google Scholar

[24]

S. P. Levandosky and W. A. Strauss, Time decay for the nonlinear beam equation,, Methods and Applications of Analysis, 7 (2000), 479. doi: Zbl 1212.35476. Google Scholar

[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. Google Scholar

[26]

G. Lebeau, Nonlinear optics and supercritical wave equation,, Bull. Soc. R. Sci. Li\`ege, 70 (2001), 267. Google Scholar

[27]

G. Lebeau, Perte de régularité pour l'équation des ondes surcritique,, Bull. Soc. Math. France, 133 (2005), 145. Google Scholar

[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. Google Scholar

[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. doi: 10.1515/gmj.2010.026. Google Scholar

[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. doi: 10.4208/jpde.v24.n4.7. Google Scholar

[31]

M. Majdoub and T. Saanouni, Global well-posedness of some critical fourth-order wave and Schrödinger equation,, preprint., (). Google Scholar

[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 }., (). Google Scholar

[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}., (). Google Scholar

[34]

J. Moser, A sharp form of an inequality of N. Trudinger,, Ind. Univ. Math. J., 20 (1971), 1077. Google Scholar

[35]

M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, Journal of Functional Analysis, 155 (1998), 364. doi: 10.1006/jfan.1997.3236. Google Scholar

[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. doi: 10.1007/PL00004737. Google Scholar

[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. Google Scholar

[38]

T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012. Google Scholar

[39]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,, Dynamics of PDE, 4 (2007), 197. doi: 01/2007.4:197-225. Google Scholar

[40]

B. Pausader, The cubic fourth-order Schrödinger equation,, Journal of Functional Analysis, 256 (2009), 2473. doi: 10.1016/j.jfa.2008.11.009. Google Scholar

[41]

B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations,, J. Differential Equations, 241 (2007), 237. doi: 10.1016/j.jde.2007.06.001. Google Scholar

[42]

B. Pausader and W. Strauss, Analyticity of the scattering operator for the beam equation,, Discrete Contin. Dyn. Syst., 25 (2009), 617. doi: 10.3934/dcds.2009.25.617. Google Scholar

[43]

L. Peletier and W. C. Troy, Higher order models in Physics and Mechanics,, Prog in Non Diff Eq and App, 45 (2001). Google Scholar

[44]

B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $R^2$,, J. Funct. Analysis, 219 (2004), 340. doi: 10.1016/j.jfa.2004.06.013. Google Scholar

[45]

B. Ruf and S. Sani, Sharp Adams-type inequalities in $R^n$,, Trans. Amer. Math. Soc., 365 (2013), 645. doi: 10.1090/S0002-9947-2012-05561-9. Google Scholar

[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. Google Scholar

[47]

T. Saanouni, Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth,, Bull. Belg. Math. Soc., 17 (2010), 441. Google Scholar

[48]

T. Saanouni, Decay of Solutions to a $2D$ Schrödinger Equation,, J. Part. Diff. Eq., 24 (2011), 37. doi: 10.4208/jpde.v24.n1.3. Google Scholar

[49]

T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space,, Math. Meth. Appl. Sci, 33 (2010), 1046. doi: 10.1002/mma.1237. Google Scholar

[50]

T. Saanouni, Remarks on the semilinear Schrödinger equation,, J. Math. Anal. Appl., 400 (2013), 331. doi: 10.1016/j.jmaa.2012.11.037. Google Scholar

[51]

I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction,, Bull. Soc. Math. France, 91 (1963), 129. Google Scholar

[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. doi: 10.1016/j.na.2006.06.020. Google Scholar

[53]

J. Shatah et M. Struwe, Well-posedness in the energy space for semilinear wave equation with critical growth,, IMRN, 7 (1994), 303. doi: 10.1155/S1073792894000346. Google Scholar

[54]

W. A. Strauss, On weak solutions of semi-linear hyperbolic equations,, Anais Acad. Brasil. Cienc., 42 (1970), 645. Google Scholar

[55]

M. Struwe, Semilinear wave equations,, Bull. Amer. Math. Soc, 26 (1992), 53. Google Scholar

[56]

M. Struwe, The critical nonlinear wave equation in 2 space dimensions,, J. European Math. Soc. (to appear)., (). Google Scholar

[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. doi: 10.1007/s00208-010-0567-6. Google Scholar

[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. Google Scholar

[59]

N. S. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar

[60]

M. Visan, The defocusing energy-critical nolinear Schrödinger equation in higher dimensions,, Duke. Math. J., 138 (2007), 281. doi: 10.1215/S0012-7094-07-13825-0. Google Scholar

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. doi: 10.1090/S0002-9939-99-05180-1. Google Scholar

[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. doi: MR204459. Google Scholar

[3]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case,, J. Amer. Math. Soc., 12 (1999), 145. doi: 10.1090/S0894-0347-99-00283-0. Google Scholar

[4]

T. Cazenave, An introduction to nonlinear Schrödinger equations,, Textos de Metodos Matematicos, 26 (1996). Google Scholar

[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. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[6]

J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ, 6 (2009), 549. doi: 10.1142/S0219891609001927. Google Scholar

[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. doi: 10.4007/annals.2008.167.767. Google Scholar

[8]

G. M. Constantine and T. H. Savitis, A multivariate Faa Di Bruno formula with applications,, T. A. M. S, 348 (1996), 503. Google Scholar

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates,, A. M. S, 120 (1998), 955. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[10]

V. A. Galaktionov and S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations,, Nonlinear Analysis, 53 (2003), 453. doi: 10.1016/S0362-546X(02)00311-5. Google Scholar

[11]

J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation,, Math. Z, 189 (1985), 487. doi: 10.1007/BF01168155. Google Scholar

[12]

M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity,, Annal. of Math., 132 (1990), 485. doi: 10.2307/1971427. Google Scholar

[13]

E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations,, (2008)., (2008). Google Scholar

[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. doi: 10.1002/cpa.20127. Google Scholar

[15]

S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves,, C. R. Acad. Sci. Paris, 345 (2007), 133. doi: 10.1016/j.crma.2007.06.008. Google Scholar

[16]

S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the $2D$ energy critical wave equation,, Duke Math., 150 (2009), 287. doi: 10.1215/00127094-2009-053. Google Scholar

[17]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion fourth-order nonlinear Schrödinger equations,, Phys. Rev. E, 53 (1996), 1336. doi: 10.1103/PhysRevE.53.R1336. Google Scholar

[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. doi: 10.1016/S0167-2789(00)00078-6. Google Scholar

[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. doi: 10.1007/s11511-008-0031-6. Google Scholar

[20]

J. Kim, A. Arnold and X. Yao, Global estimates of fundamental solutions for higher-order Schrödinger equations,, \arXiv{0807.0690v2}., (). Google Scholar

[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}, (). Google Scholar

[22]

S. P. Levandosky, Stability and instability of fourth-order solitary waves,, J. Dynam. Differential Equations, 10 (1998), 151. doi: 1040-7294/98/0100-0151S15.00/0. Google Scholar

[23]

S. P. Levandosky, Deacy estimates for fourth-order wave equations,, J. Differential Equations, 143 (1998), 360. doi: 10.1006/jdeq.1997.3369. Google Scholar

[24]

S. P. Levandosky and W. A. Strauss, Time decay for the nonlinear beam equation,, Methods and Applications of Analysis, 7 (2000), 479. doi: Zbl 1212.35476. Google Scholar

[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. Google Scholar

[26]

G. Lebeau, Nonlinear optics and supercritical wave equation,, Bull. Soc. R. Sci. Li\`ege, 70 (2001), 267. Google Scholar

[27]

G. Lebeau, Perte de régularité pour l'équation des ondes surcritique,, Bull. Soc. Math. France, 133 (2005), 145. Google Scholar

[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. Google Scholar

[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. doi: 10.1515/gmj.2010.026. Google Scholar

[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. doi: 10.4208/jpde.v24.n4.7. Google Scholar

[31]

M. Majdoub and T. Saanouni, Global well-posedness of some critical fourth-order wave and Schrödinger equation,, preprint., (). Google Scholar

[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 }., (). Google Scholar

[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}., (). Google Scholar

[34]

J. Moser, A sharp form of an inequality of N. Trudinger,, Ind. Univ. Math. J., 20 (1971), 1077. Google Scholar

[35]

M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, Journal of Functional Analysis, 155 (1998), 364. doi: 10.1006/jfan.1997.3236. Google Scholar

[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. doi: 10.1007/PL00004737. Google Scholar

[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. Google Scholar

[38]

T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012. Google Scholar

[39]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,, Dynamics of PDE, 4 (2007), 197. doi: 01/2007.4:197-225. Google Scholar

[40]

B. Pausader, The cubic fourth-order Schrödinger equation,, Journal of Functional Analysis, 256 (2009), 2473. doi: 10.1016/j.jfa.2008.11.009. Google Scholar

[41]

B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations,, J. Differential Equations, 241 (2007), 237. doi: 10.1016/j.jde.2007.06.001. Google Scholar

[42]

B. Pausader and W. Strauss, Analyticity of the scattering operator for the beam equation,, Discrete Contin. Dyn. Syst., 25 (2009), 617. doi: 10.3934/dcds.2009.25.617. Google Scholar

[43]

L. Peletier and W. C. Troy, Higher order models in Physics and Mechanics,, Prog in Non Diff Eq and App, 45 (2001). Google Scholar

[44]

B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $R^2$,, J. Funct. Analysis, 219 (2004), 340. doi: 10.1016/j.jfa.2004.06.013. Google Scholar

[45]

B. Ruf and S. Sani, Sharp Adams-type inequalities in $R^n$,, Trans. Amer. Math. Soc., 365 (2013), 645. doi: 10.1090/S0002-9947-2012-05561-9. Google Scholar

[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. Google Scholar

[47]

T. Saanouni, Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth,, Bull. Belg. Math. Soc., 17 (2010), 441. Google Scholar

[48]

T. Saanouni, Decay of Solutions to a $2D$ Schrödinger Equation,, J. Part. Diff. Eq., 24 (2011), 37. doi: 10.4208/jpde.v24.n1.3. Google Scholar

[49]

T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space,, Math. Meth. Appl. Sci, 33 (2010), 1046. doi: 10.1002/mma.1237. Google Scholar

[50]

T. Saanouni, Remarks on the semilinear Schrödinger equation,, J. Math. Anal. Appl., 400 (2013), 331. doi: 10.1016/j.jmaa.2012.11.037. Google Scholar

[51]

I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction,, Bull. Soc. Math. France, 91 (1963), 129. Google Scholar

[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. doi: 10.1016/j.na.2006.06.020. Google Scholar

[53]

J. Shatah et M. Struwe, Well-posedness in the energy space for semilinear wave equation with critical growth,, IMRN, 7 (1994), 303. doi: 10.1155/S1073792894000346. Google Scholar

[54]

W. A. Strauss, On weak solutions of semi-linear hyperbolic equations,, Anais Acad. Brasil. Cienc., 42 (1970), 645. Google Scholar

[55]

M. Struwe, Semilinear wave equations,, Bull. Amer. Math. Soc, 26 (1992), 53. Google Scholar

[56]

M. Struwe, The critical nonlinear wave equation in 2 space dimensions,, J. European Math. Soc. (to appear)., (). Google Scholar

[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. doi: 10.1007/s00208-010-0567-6. Google Scholar

[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. Google Scholar

[59]

N. S. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar

[60]

M. Visan, The defocusing energy-critical nolinear Schrödinger equation in higher dimensions,, Duke. Math. J., 138 (2007), 281. doi: 10.1215/S0012-7094-07-13825-0. Google Scholar

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