Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 995–1058 | Cite as

Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

  • Nejla Nouaili
  • Hatem Zaag


We construct a solution for the Complex Ginzburg–Landau equation in a critical case which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows us to prove the stability of the constructed solution.


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  1. 1.
    Aranson I.S., Kramer L.: The world of the complex Ginzburg–Landau equation. Rev. Modern Phys. 74(1), 99–143 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bricmont J., Kupiainen A.: Universality in blow-up for nonlinear heat equations. Nonlinearity 7(2), 539–575 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cazenave, T.: Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003Google Scholar
  4. 4.
    Côte R., Zaag H.: Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension. Commun. Pure Appl. Math. 66(10), 1541–1581 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ebde M.A., Zaag H.: Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term. SeMA J. 55, 5–21 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Giga Y., Kohn R.V.: Nondegeneracy of blowup for semilinear heat equations. Commun. Pure Appl. Math. 42(6), 845–884 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg–Landau equation. Differ. Equ. Asymptot. Anal. Math. Phys. 100, 138–152 (1996)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg–Landau equation. II. Contraction methods. Commun. Math. Phys. 187(1), 45–79 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hocking L.M., Stewartson K.: On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. Lond. Ser. A 326, 289–313 (1972)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hocking L.M., Stewartson K., Stuart J.T., Brown S.N.: A nonlinear instability in plane parallel flow. J. Fluid Mech. 51, 705–735 (1972)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Kolodner, P., Bensimon, D., Surko, M.: Traveling wave convection in an annulus. Phys. Rev. Lett. 60, 1723–1988Google Scholar
  12. 12.
    Kolodner P., Slimani S., Aubry N., Lima R.: Characterization of dispersive chaos and related states of binary-fluid convection. Phys. D 85(1-2), 165–224 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Merle F.: Solution of a nonlinear heat equation with arbitrary given blow-up points. Commun. Pure Appl. Math. 45(3), 263–300 (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Merle F., Zaag H.: Stability of the blow-up profile for equations of the type \({u_t={\Delta} u+\vert u\vert ^{p-1}u}\). Duke Math. J. 86(1), 143–195 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Masmoudi N., Zaag H.: Blow-up profile for the complex Ginzburg–Landau equation. J. Funct. Anal. 225, 1613–1666 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Nouaili N., Zaag H.: A liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up. Trans. Am. Math. Soc. 362(7), 3391–3434 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Nguyen, V.T., Zaag, H.: Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, 2015 (Submitted 2015)Google Scholar
  18. 18.
    Nouaili, N., Zaag, H.: Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation. Commun. Partial Differ. Equ. (7), 1197–1217, 2015Google Scholar
  19. 19.
    Plecháč P., Šverák V.: On self-similar singular solutions of the complex Ginzburg–Landau equation. Commun. Pure Appl. Math. 54(10), 1215–1242 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Popp S., Stiller O., Kuznetsov E., Kramer L.: The cubic complex Ginzburg–Landau equation for a backward bifurcation. Phys. D 114(1–2), 81–107 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Rottschäfer V.: Multi-bump, self-similar, blow-up solutions of the Ginzburg–Landau equation. Phys. D 237(4), 510–539 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Rottschäfer V.: Asymptotic analysis of a new type of multi-bump, self-similar, blowup solutions of the Ginzburg–Landau equation. Eur. J. Appl. Math. 24(1), 103–129 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Raphaël P., Schweyer R.: Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow. Commun. Pure Appl. Math. 66(3), 414–480 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Raphaël P., Schweyer R.: On the stability of critical chemotactic aggregation. Math. Ann. 359(1–2), 267–377 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Stewartson K., Stuart J.T.: A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545 (1971)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Tayachi, S., Zaag, H.: Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term. 2015. arXiv preprint: arXiv:1506.08306, 2015
  27. 27.
    Velázquez J.J.L.: Higher-dimensional blow up for semilinear parabolic equations. Commun. Partial Differ. Equ. 17(9-10), 1567–1596 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Velázquez J.J.L.: Classification of singularities for blowing up solutions in higher dimensions. Trans. Am. Math. Soc. 338(1), 441–464 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Velázquez J.J.L.: Estimates on the (n−1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J. 42(2), 445–476 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Velázquez, J.J.L., Galaktionov, V.A., Herrero, M.A.: The space structure near a blow-up point for semilinear heat equations: a formal approach. Zh. Vychisl. Mat. i Mat. Fiz. 31(3), 1991Google Scholar
  31. 31.
    Zaag H.: Blow-up results for vector-valued nonlinear heat equations with no gradient structure. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(5), 581– (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Zaag H.: A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure. Commun. Pure Appl. Math. 54(1), 107–133 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Zaag H.: On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(5), 505–542 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Zaag H.: One-dimensional behavior of singular N-dimensional solutions of semilinear heat equations. Commun. Math. Phys. 225(3), 523–549 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Zaag, H.: Regularity of the blow-up set and singular behavior for semilinear heat equations. In Mathematics & Mathematics Education (Bethlehem, 2000), pp. 337–347. World Science Publishing, River Edge, 2002Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CEREMADE, UMR 7534Université Paris IX-Dauphine, PSL Research UniversityParisFrance
  2. 2.LAGA, CNRS (UMR 7539)Université Paris 13, Sorbonne Paris CitéVilletaneuseFrance

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