Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

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.

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Correspondence to Thierry Cazenave.

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.

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Cazenave, T., Dias, J. & 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

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Mathematics Subject Classification (2010)

  • 35Q56
  • 35B44

Keywords

  • Complex Ginzburg–Landau equation
  • damping
  • finite-time blowup
  • energy
  • variance