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Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $
Local smooth solutions of the nonlinear Klein-gordon equation
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 |
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.
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.
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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. |
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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.
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[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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