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Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation | | Yvan Martel
; Frank Merle
; | | Date: |
12 May 2004 | | Journal: | Ann. of Math. (2), Vol. 155 (2002), no. 1, 235--280 | | Subject: | Analysis of PDEs | math.AP | | Abstract: | The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H^1(L^2 norm and energy). We consider in this paper the {it critical} generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H^1 uniform in time for all H^1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)|_{H^1} o +infty as Tuparrow T, where Tle +infty (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L^2 mass close to the minimal mass allowing blow up and with decay in L^2 at the right, we prove after rescaling and translation which leave invariant the L^2 norm that the solution converges to a {it universal} profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions. | | Source: | arXiv, math.AP/0405229 | | Services: | Forum | Review | PDF | Favorites | |
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