Blowup and ill-posedness results for a Dirac equation without gauge invariance

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June 2016, 5(2): 225-234. doi: 10.3934/eect.2016002

Blowup and ill-posedness results for a Dirac equation without gauge invariance

Piero D'Ancona ^1, and Mamoru Okamoto ^2,

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

Received January 2016 Revised April 2016 Published June 2016

Abstract
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We consider the Cauchy problem for a nonlinear Dirac equation on $\mathbb{R}^{n}$, $n\ge1$, with a power type, non gauge invariant nonlinearity $\sim|u|^{p}$. We prove several ill-posedness and blowup results for both large and small $H^{s}$ data. In particular we prove that: for (essentially arbitrary) large data in $H^{\frac n2+}(\mathbb{R} ^n)$ the solution blows up in a finite time; for suitable large $H^{s}(\mathbb{R} ^n)$ data and $s< \frac{n}{2}-\frac{1}{p-1}$ no weak solution exist; when $1< p <1+\frac1n$ (or $1< p <1+\frac2n$ in $n=1,2,3$), there exist arbitrarily small initial data data for which the solution blows up in a finite time.

Keywords: blow up, Dirac equation, non gauge invariance, $H^s$-solution., nonexistence of solution.

Mathematics Subject Classification: Primary: 35Q41; Secondary: 35B4.

Citation: Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations and Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002

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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