Local smooth solutions of the nonlinear Klein-gordon equation

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May 2021, 14(5): 1649-1672. doi: 10.3934/dcdss.2020448

Local smooth solutions of the nonlinear Klein-gordon equation

Thierry Cazenave ^1, and Ivan Naumkin ^2,,

1.	Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
2.	Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-126, Ciudad de México, 01000, México

^* Corresponding author: Ivan Naumkin

Received August 2019 Revised May 2020 Published May 2021 Early access November 2020

Fund Project: Ivan Naumkin is a Fellow of Sistema Nacional de Investigadores. The research was partially supported by project PAPIIT IA101820

Given any $ \mu _1, \mu _2\in {\mathbb C} $ and $ \alpha >0 $, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $ \partial _{ tt } u - \Delta u + \mu _1 u = \mu _2 |u|^\alpha u $ on $ {\mathbb R}^N $, $ N\ge 1 $, that do not vanish, i.e. $ |u (t, x) | >0 $ for all $ x \in {\mathbb R}^N $ and all sufficiently small $ t $. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.

Keywords: Nonlinear Klein-Gordon equation, nonlinear wave equation, nonlinear Dirac equation, local existence, smooth solutions, non-vanishing solutions.

Mathematics Subject Classification: Primary: 35L70; secondary: 35L60, 35A01, 35B65.

Citation: Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev spaces. Second edition, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2003.
[2]	I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^1(\Bbb{R}^3)$, Commun. Math. Phys., 335 (2015), 43-82. doi: 10.1007/s00220-014-2164-0.
[3]	I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^{\frac 12}(\Bbb R^2)$, Commun. Math. Phys., 343 (2016), 515-562. doi: 10.1007/s00220-015-2508-4.
[4]	P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186 (1984), 383-391. doi: 10.1007/BF01174891.
[5]	P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344. doi: 10.1016/0022-0396(85)90083-X.
[6]	T. Candy, Global existence for an $L^{2}$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.
[7]	T. Candy and H. Lindblad, Long range scattering for the cubic Dirac equation on $\Bbb R^{1+1}$, Differential Integral Equations, 31 (2018), 507-518.
[8]	T. Cazenave, F. Dickstein and F. B. Weissler, Non-regularity in Hölder and Sobolev spaces of solutions to the semilinear heat and Schrödinger equations, Nagoya Math. J., 226 (2017), 44-70. doi: 10.1017/nmj.2016.35.
[9]	T. Cazenave and I. Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math., 19 (2017), 1650038, 20 pp. doi: 10.1142/S0219199716500383.
[10]	T. Cazenave and I. Naumkin, Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal., 274 (2018), 402-432. doi: 10.1016/j.jfa.2017.10.022.
[11]	J.-M. Delort, Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4) 34 (2001), 1–61. doi: 10.1016/S0012-9593(00)01059-4.
[12]	J.-M. Delort, D. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323. doi: 10.1016/j.jfa.2004.01.008.
[13]	J.-P. Dias and M. Figueira, Time decay for the solutions of a nonlinear Dirac equation in one space dimension, Ricerche Mat., 35 (1986), 309-316.
[14]	M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s({\bf{R}}^3)$ for $s>1$, SIAM J. Math. Anal., 28 (1997), 338-362. doi: 10.1137/S0036141095283017.
[15]	J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.
[16]	J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 399-442.
[17]	J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15-35. doi: 10.1016/S0294-1449(16)30329-8.
[18]	J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573. doi: 10.1007/BF01218585.
[19]	N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028. doi: 10.1007/s00033-007-7008-8.
[20]	N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations, 244 (2008), 188-199. doi: 10.1016/j.jde.2007.10.002.
[21]	N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math., 11 (2009), 771-781. doi: 10.1142/S0219199709003582.
[22]	H. Kalf and O. Yamada, Essential self-adjointness of $n$-dimensional Dirac operators with a variable mass term, J. Math. Phys., 42 (2001), 2667-2676. doi: 10.1063/1.1367331.
[23]	S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213. doi: 10.1215/kjm/1250517908.
[24]	S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641. doi: 10.1002/cpa.3160380512.
[25]	F. Linares, H. Miyazaki and G. Ponce, On a class of solutions to the generalized KdV type equation, Commun. Contemp. Math., 21 (2019), 1850056, 21 pp. doi: 10.1142/S0219199718500566.
[26]	F. Linares, G. Ponce and G. N. Santos, On a class of solutions to the generalized derivative Schrödinger equations, Acta. Math. Sin. (Engl. Ser.), 35 (2019), 1057-1073. doi: 10.1007/s10114-019-7540-4.
[27]	F. Linares, G. Ponce and G. N. Santos, On a class to the generalized derivative Schrödinger equations II, J. Differential Equations, 267 (2019), 97-118. doi: 10.1016/j.jde.2019.01.004.
[28]	H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258. doi: 10.1007/s11005-005-0021-y.
[29]	S. Machihara, One dimensional Dirac equation with quadratic nonlinearities, Discrete Contin. Dyn. Syst., 13 (2005), 277-290. doi: 10.3934/dcds.2005.13.277.
[30]	S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math., 9 (2007), 421-435. doi: 10.1142/S0219199707002484.
[31]	S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal., 219 (2005), 1-20. doi: 10.1016/j.jfa.2004.07.005.
[32]	S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the relativistic limit for the nonlinear Dirac equation, Rev. Math. Iberoamericana, 19 (2003), 179-194. doi: 10.4171/RMI/342.
[33]	S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math., 50 (2010), 403-451. doi: 10.1215/0023608X-2009-018.
[34]	S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, Trans. Amer. Math. Soc., 370 (2018), 8155-8170. doi: 10.1090/tran/7262.
[35]	C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 1-31. doi: 10.1002/cpa.3160250103.
[36]	K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520.
[37]	K. Moriyama, S. Tonegawa and Y. Tsutsumi, Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension, Funkcial. Ekvac., 40 (1997), 313-333.
[38]	K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.
[39]	I. P. Naumkin, Cubic nonlinear Dirac equation in a quarter plane, J. Math. Anal. Appl., 434 (2016), 1633-1664. doi: 10.1016/j.jmaa.2015.09.049.
[40]	I. P. Naumkin, Klein-Gordon equation with critical nonlinearity and inhomogeneous Dirichlet boundary conditions, Differential Integral Equations, 29 (2016), 55-92.
[41]	I. P. Naumkin, Initial-boundary value problem for the one dimensional Thirring model, J. Differential Equations, 261 (2016), 4486-4523. doi: 10.1016/j.jde.2016.07.003.
[42]	I. Naumkin, Neumann problem for the nonlinear Klein-Gordon equation, Nonlinear Anal., 149 (2017), 81-110. doi: 10.1016/j.na.2016.10.014.
[43]	T. Ozawa, K. Tsutaya and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z., 222 (1996), 341-362. doi: 10.1007/BF02621870.
[44]	H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270. doi: 10.1007/BF01181697.
[45]	H. Pecher, Low-energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal., 63 (1985), 101-122. doi: 10.1016/0022-1236(85)90100-4.
[46]	H. Pecher, Local solutions of semilinear wave equations in $H^{s+1}$, Math. Methods Appl. Sci., 19 (1996), 145-170. doi: 10.1002/(SICI)1099-1476(19960125)19:2<145::AID-MMA767>3.0.CO;2-M.
[47]	H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 673-685. doi: 10.3934/cpaa.2014.13.673.
[48]	H. Sasaki, Small data scattering for the one-dimensional nonlinear Dirac equation with power nonlinearity, Comm. Partial Differential Equations, 40 (2015), 1959-2004. doi: 10.1080/03605302.2015.1081608.
[49]	S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations, 23 (2010), 265-278.
[50]	J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.
[51]	M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769. doi: 10.1103/PhysRevD.1.2766.
[52]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
[53]	W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. doi: 10.1016/0022-1236(81)90063-X.
[54]	W. A. Strauss, Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293. doi: 10.1016/0022-1236(81)90019-7.
[55]	B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.
[56]	W. E. Thirring, A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112. doi: 10.1016/0003-4916(58)90015-0.
[57]	N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22 (1998), 193-211. doi: 10.21099/tkbjm/1496163480.

show all references

References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev spaces. Second edition, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2003.
[2]	I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^1(\Bbb{R}^3)$, Commun. Math. Phys., 335 (2015), 43-82. doi: 10.1007/s00220-014-2164-0.
[3]	I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^{\frac 12}(\Bbb R^2)$, Commun. Math. Phys., 343 (2016), 515-562. doi: 10.1007/s00220-015-2508-4.
[4]	P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186 (1984), 383-391. doi: 10.1007/BF01174891.
[5]	P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344. doi: 10.1016/0022-0396(85)90083-X.
[6]	T. Candy, Global existence for an $L^{2}$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.
[7]	T. Candy and H. Lindblad, Long range scattering for the cubic Dirac equation on $\Bbb R^{1+1}$, Differential Integral Equations, 31 (2018), 507-518.
[8]	T. Cazenave, F. Dickstein and F. B. Weissler, Non-regularity in Hölder and Sobolev spaces of solutions to the semilinear heat and Schrödinger equations, Nagoya Math. J., 226 (2017), 44-70. doi: 10.1017/nmj.2016.35.
[9]	T. Cazenave and I. Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math., 19 (2017), 1650038, 20 pp. doi: 10.1142/S0219199716500383.
[10]	T. Cazenave and I. Naumkin, Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal., 274 (2018), 402-432. doi: 10.1016/j.jfa.2017.10.022.
[11]	J.-M. Delort, Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4) 34 (2001), 1–61. doi: 10.1016/S0012-9593(00)01059-4.
[12]	J.-M. Delort, D. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323. doi: 10.1016/j.jfa.2004.01.008.
[13]	J.-P. Dias and M. Figueira, Time decay for the solutions of a nonlinear Dirac equation in one space dimension, Ricerche Mat., 35 (1986), 309-316.
[14]	M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s({\bf{R}}^3)$ for $s>1$, SIAM J. Math. Anal., 28 (1997), 338-362. doi: 10.1137/S0036141095283017.
[15]	J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.
[16]	J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 399-442.
[17]	J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15-35. doi: 10.1016/S0294-1449(16)30329-8.
[18]	J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573. doi: 10.1007/BF01218585.
[19]	N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028. doi: 10.1007/s00033-007-7008-8.
[20]	N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations, 244 (2008), 188-199. doi: 10.1016/j.jde.2007.10.002.
[21]	N. Hayashi and P. I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math., 11 (2009), 771-781. doi: 10.1142/S0219199709003582.
[22]	H. Kalf and O. Yamada, Essential self-adjointness of $n$-dimensional Dirac operators with a variable mass term, J. Math. Phys., 42 (2001), 2667-2676. doi: 10.1063/1.1367331.
[23]	S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213. doi: 10.1215/kjm/1250517908.
[24]	S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641. doi: 10.1002/cpa.3160380512.
[25]	F. Linares, H. Miyazaki and G. Ponce, On a class of solutions to the generalized KdV type equation, Commun. Contemp. Math., 21 (2019), 1850056, 21 pp. doi: 10.1142/S0219199718500566.
[26]	F. Linares, G. Ponce and G. N. Santos, On a class of solutions to the generalized derivative Schrödinger equations, Acta. Math. Sin. (Engl. Ser.), 35 (2019), 1057-1073. doi: 10.1007/s10114-019-7540-4.
[27]	F. Linares, G. Ponce and G. N. Santos, On a class to the generalized derivative Schrödinger equations II, J. Differential Equations, 267 (2019), 97-118. doi: 10.1016/j.jde.2019.01.004.
[28]	H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258. doi: 10.1007/s11005-005-0021-y.
[29]	S. Machihara, One dimensional Dirac equation with quadratic nonlinearities, Discrete Contin. Dyn. Syst., 13 (2005), 277-290. doi: 10.3934/dcds.2005.13.277.
[30]	S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math., 9 (2007), 421-435. doi: 10.1142/S0219199707002484.
[31]	S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal., 219 (2005), 1-20. doi: 10.1016/j.jfa.2004.07.005.
[32]	S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the relativistic limit for the nonlinear Dirac equation, Rev. Math. Iberoamericana, 19 (2003), 179-194. doi: 10.4171/RMI/342.
[33]	S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math., 50 (2010), 403-451. doi: 10.1215/0023608X-2009-018.
[34]	S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, Trans. Amer. Math. Soc., 370 (2018), 8155-8170. doi: 10.1090/tran/7262.
[35]	C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 1-31. doi: 10.1002/cpa.3160250103.
[36]	K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520.
[37]	K. Moriyama, S. Tonegawa and Y. Tsutsumi, Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension, Funkcial. Ekvac., 40 (1997), 313-333.
[38]	K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.
[39]	I. P. Naumkin, Cubic nonlinear Dirac equation in a quarter plane, J. Math. Anal. Appl., 434 (2016), 1633-1664. doi: 10.1016/j.jmaa.2015.09.049.
[40]	I. P. Naumkin, Klein-Gordon equation with critical nonlinearity and inhomogeneous Dirichlet boundary conditions, Differential Integral Equations, 29 (2016), 55-92.
[41]	I. P. Naumkin, Initial-boundary value problem for the one dimensional Thirring model, J. Differential Equations, 261 (2016), 4486-4523. doi: 10.1016/j.jde.2016.07.003.
[42]	I. Naumkin, Neumann problem for the nonlinear Klein-Gordon equation, Nonlinear Anal., 149 (2017), 81-110. doi: 10.1016/j.na.2016.10.014.
[43]	T. Ozawa, K. Tsutaya and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z., 222 (1996), 341-362. doi: 10.1007/BF02621870.
[44]	H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270. doi: 10.1007/BF01181697.
[45]	H. Pecher, Low-energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal., 63 (1985), 101-122. doi: 10.1016/0022-1236(85)90100-4.
[46]	H. Pecher, Local solutions of semilinear wave equations in $H^{s+1}$, Math. Methods Appl. Sci., 19 (1996), 145-170. doi: 10.1002/(SICI)1099-1476(19960125)19:2<145::AID-MMA767>3.0.CO;2-M.
[47]	H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 673-685. doi: 10.3934/cpaa.2014.13.673.
[48]	H. Sasaki, Small data scattering for the one-dimensional nonlinear Dirac equation with power nonlinearity, Comm. Partial Differential Equations, 40 (2015), 1959-2004. doi: 10.1080/03605302.2015.1081608.
[49]	S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations, 23 (2010), 265-278.
[50]	J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.
[51]	M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769. doi: 10.1103/PhysRevD.1.2766.
[52]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
[53]	W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. doi: 10.1016/0022-1236(81)90063-X.
[54]	W. A. Strauss, Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293. doi: 10.1016/0022-1236(81)90019-7.
[55]	B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.
[56]	W. E. Thirring, A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112. doi: 10.1016/0003-4916(58)90015-0.
[57]	N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22 (1998), 193-211. doi: 10.21099/tkbjm/1496163480.

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