×

Blowup and ill-posedness results for a Dirac equation without gauge invariance. (English) Zbl 1346.35172

Summary: We consider the Cauchy problem for a nonlinear Dirac equation on \(\mathbb{R}^{n}\), \(n\geq 1\), 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 \(H^{\frac{n}{2}+}(\mathbb{R}^n)\) 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 \(1< p <1+\frac{1}{n}\) (or \(1< p <1+\frac{2}{n}\) in \(n=1,2,3\)), there exist arbitrarily small initial data data for which the solution blows up in a finite time.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35B44 Blow-up in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. Bejenaru, The cubic Dirac equation: Small initial data in \(H^1(\mathbbR^3)\),, Comm. Math. Phys., 335, 43 (2015) · Zbl 1321.35180 · doi:10.1007/s00220-014-2164-0
[2] I. Bejenaru, The cubic Dirac equation: Small initial data in \(H^{1/2}(\mathbbR^2)\),, Comm. Math. Phys., 343, 515 (2016) · Zbl 1339.35261 · doi:10.1007/s00220-015-2508-4
[3] N. Bournaveas, Global well-posedness for the massless cubic Dirac equation,, Int Math Res Notices in press. · Zbl 0953.35003 · 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, 643 (2011) · Zbl 1229.35225
[5] T. Cazenave, <em>Semilinear Schrödinger Equations</em>,, Courant Lect. Notes Math. (2003) · Zbl 1055.35003
[6] M. Escobedo, A semilinear Dirac equation in \(H^s(\mathbbR^3)\) for \(s>1\),, SIAM J. Math. Anal., 28, 338 (1997) · Zbl 0877.35028 · doi:10.1137/S0036141095283017
[7] R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations,, Math. Z., 177, 323 (1981) · Zbl 0438.35045 · doi:10.1007/BF01162066
[8] M. Ikeda, Small-data blow-up of \(L^2\)-solution for the nonlinear Schrödinger equation without gauge invariance,, Differential Integral Equations, 26, 1275 (2013) · Zbl 1313.35324
[9] M. Ikeda, Small data blow-up of \(L^2\) or \(H^1\)-solution for the semilinear Schrödinger equation without gauge invariance,, J. Evol. Equ., 15, 571 (2015) · Zbl 1327.35353 · doi:10.1007/s00028-015-0273-7
[10] M. Ikeda, Some non-existence results for the semilinear Schrödinger equation without gauge invariance,, J. Math. Anal. Appl., 425, 758 (2015) · Zbl 1311.35288 · 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, 235 (1979) · Zbl 0406.35042 · doi:10.1007/BF01647974
[12] S. Machihara, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation,, J. Funct. Anal., 219, 1 (2005) · Zbl 1060.35025 · 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, 389 (2012) · Zbl 1239.35149 · 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, 673 (2014) · Zbl 1296.35174 · doi:10.3934/cpaa.2014.13.673
[15] T. Runst, <em>Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations</em>,, de Gruyter Series in Nonlinear Analysis and Applications, 3 (1996) · Zbl 0873.35001 · doi:10.1515/9783110812411
[16] T. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions,, J. Differential Equations, 52, 378 (1984) · Zbl 0555.35091 · doi:10.1016/0022-0396(84)90169-4
[17] Q. Zhang, Blow-up results for nonlinear parabolic equations on manifolds,, Duke Math. J., 97, 515 (1999) · Zbl 0954.35029 · 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, 333, 109 (2001) · Zbl 1056.35123 · doi:10.1016/S0764-4442(01)01999-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.