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Blow up in finite time and dynamics of blow up solutions for the  -critical generalized KdV equation
Author(s):
Yvan
Martel;
Frank
Merle
Journal:
J. Amer. Math. Soc.
15
(2002),
617-664.
MSC (1991):
Primary 35Q53;
Secondary 35B05, 35Q51
Posted:
March 8, 2002
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Abstract:
In this paper, we describe the dynamics of blow up solutions for the critical generalized KdV equation such that the initial data is close to the soliton in  and has decay in  at the right. In particular, we prove that blow up occurs in finite time, and we obtain an upper bound on the blow up rate.
References:
-
- 1.
- J.L. Bona, V.A. Dougalis, O.A. Karakashian and W.R. McKinney, Conservative, high order numerical schemes, Phil. Trans. Roy. Soc. London Ser. A. 351 (1995), 107-164. MR 96d:65141
- 2.
- J. Bourgain, Harmonic analysis and nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians, 1,2 (Zurich, 1994), 31-44, Birkhäuser, Basel, 1995. MR 97f:35088
- 3.
- J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20, 6 (1989), 1388-1425. MR 90i:35240
- 4.
- T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematical Supplementary Studies, Studies in Applied Math. 8 (1983), 93-128. MR 86f:35160
- 5.
- C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620. MR 94h:35229
- 6.
- P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. MR 38:3620
- 7.
- Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), 74-123.
- 8.
- Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. 79 (2000), 339-425. MR 2001i:37102
- 9.
- Y. Martel and F. Merle, Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation, to appear in Ann. of Math.
- 10.
- F. Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceeding of the International Congress of Mathematicians (Berlin, 1998), Doc. Math. J. DMV. MR 99h:35200
- 11.
- F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), 555-578.
- 12.
- R.M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review 18 (1976), 412-459. MR 53:8689
- 13.
- M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576. MR 84d:35140
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Additional Information:
Yvan
Martel
Affiliation:
Département de Mathématiques, Université de Cergy--Pontoise, 2, av. A. Chauvin, 95302 Cergy Pontoise, France
Email:
Yvan.Martel@math.u-cergy.fr
Frank
Merle
Affiliation:
Département de Mathématiques, Université de Cergy--Pontoise, 2, av. A. Chauvin, 95302 Cergy Pontoise, France -- and -- Institut Universitaire de France
Email:
Frank.Merle@math.u-cergy.fr
DOI:
10.1090/S0894-0347-02-00392-2
PII:
S 0894-0347(02)00392-2
Keywords:
Critical KdV equation,
finite time blow up,
blow up rate
Received by editor(s):
March 15, 2001
Posted:
March 8, 2002
Copyright of article:
Copyright
2002,
American Mathematical Society
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