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

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 & 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.
[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.
[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.
[4]	T. Cazenave, An introduction to nonlinear Schrödinger equations,, Textos de Metodos Matematicos, 26 (1996).
[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.
[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.
[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.
[8]	G. M. Constantine and T. H. Savitis, A multivariate Faa Di Bruno formula with applications,, T. A. M. S, 348 (1996), 503.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates,, A. M. S, 120 (1998), 955. 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. 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. 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. doi: 10.2307/1971427.
[13]	E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations,, (2008)., (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. doi: 10.1002/cpa.20127.
[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.
[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.
[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.
[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.
[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.
[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. 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. 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. 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.
[26]	G. Lebeau, Nonlinear optics and supercritical wave equation,, Bull. Soc. R. Sci. Li\`ege, 70 (2001), 267.
[27]	G. Lebeau, Perte de régularité pour l'équation des ondes surcritique,, Bull. Soc. Math. France, 133 (2005), 145.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. 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. doi: 01/2007.4:197-225.
[40]	B. Pausader, The cubic fourth-order Schrödinger equation,, Journal of Functional Analysis, 256 (2009), 2473. 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. 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. 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. 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. 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.
[47]	T. Saanouni, Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth,, Bull. Belg. Math. Soc., 17 (2010), 441.
[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.
[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.
[50]	T. Saanouni, Remarks on the semilinear Schrödinger equation,, J. Math. Anal. Appl., 400 (2013), 331. 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.
[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.
[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.
[54]	W. A. Strauss, On weak solutions of semi-linear hyperbolic equations,, Anais Acad. Brasil. Cienc., 42 (1970), 645.
[55]	M. Struwe, Semilinear wave equations,, Bull. Amer. Math. Soc, 26 (1992), 53.
[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. 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.
[59]	N. S. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.
[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.

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.
[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.
[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.
[4]	T. Cazenave, An introduction to nonlinear Schrödinger equations,, Textos de Metodos Matematicos, 26 (1996).
[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.
[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.
[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.
[8]	G. M. Constantine and T. H. Savitis, A multivariate Faa Di Bruno formula with applications,, T. A. M. S, 348 (1996), 503.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates,, A. M. S, 120 (1998), 955. 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. 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. 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. doi: 10.2307/1971427.
[13]	E. Hebey and B. Pausader, An introduction to fourth-order nonlinear wave equations,, (2008)., (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. doi: 10.1002/cpa.20127.
[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.
[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.
[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.
[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.
[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.
[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. 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. 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. 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.
[26]	G. Lebeau, Nonlinear optics and supercritical wave equation,, Bull. Soc. R. Sci. Li\`ege, 70 (2001), 267.
[27]	G. Lebeau, Perte de régularité pour l'équation des ondes surcritique,, Bull. Soc. Math. France, 133 (2005), 145.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. 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. doi: 01/2007.4:197-225.
[40]	B. Pausader, The cubic fourth-order Schrödinger equation,, Journal of Functional Analysis, 256 (2009), 2473. 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. 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. 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. 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. 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.
[47]	T. Saanouni, Global well-posedness and scattering of a 2D Schrödinger equation with exponential growth,, Bull. Belg. Math. Soc., 17 (2010), 441.
[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.
[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.
[50]	T. Saanouni, Remarks on the semilinear Schrödinger equation,, J. Math. Anal. Appl., 400 (2013), 331. 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.
[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.
[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.
[54]	W. A. Strauss, On weak solutions of semi-linear hyperbolic equations,, Anais Acad. Brasil. Cienc., 42 (1970), 645.
[55]	M. Struwe, Semilinear wave equations,, Bull. Amer. Math. Soc, 26 (1992), 53.
[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. 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.
[59]	N. S. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.
[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.

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