# American Institute of Mathematical Sciences

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. 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