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

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

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.
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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