Abstract
In this paper, we consider the complex Ginzburg–Landau equation \({u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}\) on \({\mathbb{R}^N}\), where \({\alpha > 0,\,\gamma \in \mathbb{R}}\) and \({-\pi /2 < \theta < \pi /2}\). By convexity arguments, we prove that, under certain conditions on \({\alpha,\theta,\gamma}\), a class of solutions with negative initial energy blows up in finite time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cazenave T., Dickstein F. and Weissler F.B. Finite-time blowup for a complex Ginzburg–Landau equation. SIAM J. Math. Anal. 45 (2013), no. 1, 244–266. (doi:10.1137/120878690)
Doering C.R., Gibbon J.D., Holm D.D., and Nicolaenko B. Low-dimensional behaviour in the complex Ginzburg–Landau equation. Nonlinearity 1 (1988), no. 2, 279–309. (doi:10.1088/0951-7715/1/2/001)
Glassey R.T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794–1797.
Levine H.A. Some nonexistence and instability theorems for formally parabolic equations of the form Pu t = −Au + f(u), Arch. Ration. Mech. Anal. 51 (1973), 371–386. (doi:10.1007/BF00263041)
Masmoudi N. and Zaag H. Blow-up profile for the complex Ginzburg–Landau equation, J. Funct. Anal. 255 (2008), no. 7, 1613–1666. (doi:10.1016/j.jfa.2008.03.008)
Mohamad D. Blow-up for the damped L 2-critical nonlinear Schrödinger equation. Adv. Differential Equations 17 (2012), no. 3-4, 337–367. (link: http://projecteuclid.org/euclid.ade/1355703089)
Ohta M. and Todorova G. Remarks on global existence and blowup for damped nonlinear Schrdinger equations. Discrete Contin. Dyn. Syst. 23 (2009), no. 4, 1313–1325. (doi:10.3934/dcds.2009.23.1313)
Quittner P. and Souplet P. Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007. (doi:10.1007/978-3-7643-8442-5)
Tsutsumi M. Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equation. SIAM J. Math. Anal. 15 (1984), no. 2, 357–366. (doi:10.1137/0515028)
Weissler F.B. Local existence and nonexistence for semilinear parabolic equations in L p, Indiana Univ. Math. J. 29 (1980), no. 1, 79–102. (link: http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1980/29/29007/pdf)
Zakharov V.E.: Collapse of Langmuir waves, Soviet Phys. JETP 35, 908–914 (1972)
Author information
Authors and Affiliations
Corresponding author
Additional information
João Paulo Dias and Mário Figueira: research partially supported by the Portuguese Foundation for Science and Technology (FCT) through the Grant PTDC/MAT/110613/2009 and by PEstOE/MAT/UI 0209/2011.
Rights and permissions
About this article
Cite this article
Cazenave, T., Dias, JP. & Figueira, M. Finite-time blowup for a complex Ginzburg–Landau equation with linear driving. J. Evol. Equ. 14, 403–415 (2014). https://doi.org/10.1007/s00028-014-0220-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-014-0220-z