Blow up for the critical gKdV equation III: exotic regimes

• Yvan Martel ^[3] ; Frank Merle ^[1] ; Pierre Raphaël ^[2]
  1. [1] Cergy-Pontoise University

     Cergy-Pontoise University

     Arrondissement de Pontoise, Francia

     
  2. [2] Paul Sabatier University

     Paul Sabatier University

     Arrondissement de Toulouse, Francia

     
  3. [3] Université de Versailles St-Quentin, France

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• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 14, Nº 2, 2015, págs. 575-631
• Idioma: inglés
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• Resumen

  • We consider the blow-up problem in H1 for the L2 critical generalized Korteweg–de Vries (gKdV) equation, as a continuation of [38, 39]. We know from [38] that the unique and stable blow-up rate for H1 solutions close to the solitons with strong decay on the right is kux (t)kL2 ⇠ 1 T − t as t " T < +1.

    In this paper we construct non-generic blow-up regimes in H1 by considering initial data with explicit slow decay on the right in space. We obtain finite time blow-up solutions with speed kux (t)kL2 ⇠ 1 (T − t)⌫ as t " T < +1, ⌫ > 11 13 , as well as global in time growing up solutions with exponential growth kux (t)kL2 ⇠ et as t !+1, or growth of any power kux (t)kL2 ⇠ t⌫ as t !+1, ⌫ > 0.

    These solutions can be taken with initial data arbitrarily close in H1 to the ground state solitary wave.