Resumen de Blow up for the critical gKdV equation III: exotic regimes

Yvan Martel, Frank Merle, Pierre Raphaël

• 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.