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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

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 & 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.  Google Scholar

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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.  Google Scholar

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T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.  Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, American Mathematical Society, 2003.  Google Scholar

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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.  Google Scholar

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R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340. doi: 10.1007/BF01162066.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations, 16 (2011), 643-666.  Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, American Mathematical Society, 2003.  Google Scholar

[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.  Google Scholar

[7]

R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340. doi: 10.1007/BF01162066.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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