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Blowup and ill-posedness results for a Dirac equation without gauge invariance
1. | Dipartimento di Matematica, Unversità di Roma "La Sapienza", Piazzale A. More 2, 00185 Roma, Italy |
2. | Department of Mathematics, Institute of Engineering, Academic Assembly, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553 |
References:
[1] |
I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^1(\mathbbR^3)$, Comm. Math. Phys., 335 (2015), 43-82.
doi: 10.1007/s00220-014-2164-0. |
[2] |
I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^{1/2}(\mathbbR^2)$, Comm. Math. Phys., 343 (2016), 515-562.
doi: 10.1007/s00220-015-2508-4. |
[3] |
N. Bournaveas and T. Candy, Global well-posedness for the massless cubic Dirac equation, Int Math Res Notices in press.
doi: 10.1093/imrn/rnv361. |
[4] |
T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, American Mathematical Society, 2003. |
[6] |
M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s(\mathbbR^3)$ for $s>1$, SIAM J. Math. Anal., 28 (1997), 338-362.
doi: 10.1137/S0036141095283017. |
[7] |
R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.
doi: 10.1007/BF01162066. |
[8] |
M. Ikeda and Y. Wakasugi, Small-data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance, Differential Integral Equations, 26 (2013), 1275-1285. |
[9] |
M. Ikeda and T. Inui, Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.
doi: 10.1007/s00028-015-0273-7. |
[10] |
M. Ikeda and T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425 (2015), 758-773.
doi: 10.1016/j.jmaa.2015.01.003. |
[11] |
F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.
doi: 10.1007/BF01647974. |
[12] |
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. |
[13] |
T. Oh, A blowup result for the periodic NLS without gauge invariance, C. R. Acad. Sci. Paris. Ser., 350 (2012), 389-392.
doi: 10.1016/j.crma.2012.04.009. |
[14] |
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. |
[15] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, 3, Berlin, 1996.
doi: 10.1515/9783110812411. |
[16] |
T. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.
doi: 10.1016/0022-0396(84)90169-4. |
[17] |
Q. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515-539.
doi: 10.1215/S0012-7094-99-09719-3. |
[18] |
Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris, Ser. I Math., 333 (2001), 109-114.
doi: 10.1016/S0764-4442(01)01999-1. |
show all references
References:
[1] |
I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^1(\mathbbR^3)$, Comm. Math. Phys., 335 (2015), 43-82.
doi: 10.1007/s00220-014-2164-0. |
[2] |
I. Bejenaru and S. Herr, The cubic Dirac equation: Small initial data in $H^{1/2}(\mathbbR^2)$, Comm. Math. Phys., 343 (2016), 515-562.
doi: 10.1007/s00220-015-2508-4. |
[3] |
N. Bournaveas and T. Candy, Global well-posedness for the massless cubic Dirac equation, Int Math Res Notices in press.
doi: 10.1093/imrn/rnv361. |
[4] |
T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, American Mathematical Society, 2003. |
[6] |
M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s(\mathbbR^3)$ for $s>1$, SIAM J. Math. Anal., 28 (1997), 338-362.
doi: 10.1137/S0036141095283017. |
[7] |
R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.
doi: 10.1007/BF01162066. |
[8] |
M. Ikeda and Y. Wakasugi, Small-data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance, Differential Integral Equations, 26 (2013), 1275-1285. |
[9] |
M. Ikeda and T. Inui, Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.
doi: 10.1007/s00028-015-0273-7. |
[10] |
M. Ikeda and T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425 (2015), 758-773.
doi: 10.1016/j.jmaa.2015.01.003. |
[11] |
F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.
doi: 10.1007/BF01647974. |
[12] |
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. |
[13] |
T. Oh, A blowup result for the periodic NLS without gauge invariance, C. R. Acad. Sci. Paris. Ser., 350 (2012), 389-392.
doi: 10.1016/j.crma.2012.04.009. |
[14] |
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. |
[15] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, 3, Berlin, 1996.
doi: 10.1515/9783110812411. |
[16] |
T. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.
doi: 10.1016/0022-0396(84)90169-4. |
[17] |
Q. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515-539.
doi: 10.1215/S0012-7094-99-09719-3. |
[18] |
Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris, Ser. I Math., 333 (2001), 109-114.
doi: 10.1016/S0764-4442(01)01999-1. |
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