## Existence of blow-up solutions in the energy space for the critical generalized KdV equation

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**14**(2001), 555-578 Request permission

## Abstract:

For the critical generalized Korteweg–de Vries equation, we establish blow-up in finite or infinite time in $H^1(\mathbf R)$ for initial data with negative energy, close to a soliton up to scaling and translation.## References

- J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney,
*Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation*, Philos. Trans. Roy. Soc. London Ser. A**351**(1995), no. 1695, 107–164. MR**1336983**, DOI 10.1098/rsta.1995.0027 - J. L. Bona, P. E. Souganidis, and W. A. Strauss,
*Stability and instability of solitary waves of Korteweg-de Vries type*, Proc. Roy. Soc. London Ser. A**411**(1987), no. 1841, 395–412. MR**897729** - Jean Bourgain,
*Harmonic analysis and nonlinear partial differential equations*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 31–44. MR**1403913** - J. Bourgain,
*Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations*, Geom. Funct. Anal.**3**(1993), no. 2, 107–156. MR**1209299**, DOI 10.1007/BF01896020 - T. Cazenave and P.-L. Lions,
*Orbital stability of standing waves for some nonlinear Schrödinger equations*, Comm. Math. Phys.**85**(1982), no. 4, 549–561. MR**677997**, DOI 10.1007/BF01403504 - B. Gidas, Wei Ming Ni, and L. Nirenberg,
*Symmetry and related properties via the maximum principle*, Comm. Math. Phys.**68**(1979), no. 3, 209–243. MR**544879**, DOI 10.1007/BF01221125 - B. Gidas and J. Spruck,
*A priori bounds for positive solutions of nonlinear elliptic equations*, Comm. Partial Differential Equations**6**(1981), no. 8, 883–901. MR**619749**, DOI 10.1080/03605308108820196 - J. Ginibre and Y. Tsutsumi,
*Uniqueness of solutions for the generalized Korteweg-de Vries equation*, SIAM J. Math. Anal.**20**(1989), no. 6, 1388–1425. MR**1019307**, DOI 10.1137/0520091 - Manoussos Grillakis, Jalal Shatah, and Walter Strauss,
*Stability theory of solitary waves in the presence of symmetry. I*, J. Funct. Anal.**74**(1987), no. 1, 160–197. MR**901236**, DOI 10.1016/0022-1236(87)90044-9 - Tosio Kato,
*On the Cauchy problem for the (generalized) Korteweg-de Vries equation*, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR**759907** - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle*, Comm. Pure Appl. Math.**46**(1993), no. 4, 527–620. MR**1211741**, DOI 10.1002/cpa.3160460405
KPV3 C.E. Kenig, G. Ponce and L. Vega, On the concentration of blow-up solutions for the generalized KdV equation critical in $L^2$, Contemp. Math. - Peter D. Lax,
*Integrals of nonlinear equations of evolution and solitary waves*, Comm. Pure Appl. Math.**21**(1968), 467–490. MR**235310**, DOI 10.1002/cpa.3160210503
MM Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg–de Vries equation, to appear in Geometrical and Functional Analysis.
MM2 Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg–de Vries equation, Journal de Math. Pures et Appliquees - Frank Merle,
*Asymptotics for $L^2$ minimal blow-up solutions of critical nonlinear Schrödinger equation*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**13**(1996), no. 5, 553–565 (English, with English and French summaries). MR**1409662**, DOI 10.1016/S0294-1449(16)30114-7 - Frank Merle,
*Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations*, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 57–66. MR**1648140**
MZ F. Merle and H. Zaag, A Liouville Theorem for a vector valued nonlinear heat equation and applications, Math. Ann. - Robert L. Pego and Michael I. Weinstein,
*Asymptotic stability of solitary waves*, Comm. Math. Phys.**164**(1994), no. 2, 305–349. MR**1289328**, DOI 10.1007/BF02101705 - Martin Schechter,
*Spectra of partial differential operators*, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR**869254** - A. Soffer and M. I. Weinstein,
*Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations*, Invent. Math.**136**(1999), no. 1, 9–74. MR**1681113**, DOI 10.1007/s002220050303 - A. R. Collar,
*On the reciprocation of certain matrices*, Proc. Roy. Soc. Edinburgh**59**(1939), 195–206. MR**8**, DOI 10.1017/S0370164600012281 - Michael I. Weinstein,
*Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys.**87**(1982/83), no. 4, 567–576. MR**691044**, DOI 10.1007/BF01208265 - Michael I. Weinstein,
*Modulational stability of ground states of nonlinear Schrödinger equations*, SIAM J. Math. Anal.**16**(1985), no. 3, 472–491. MR**783974**, DOI 10.1137/0516034 - Michael I. Weinstein,
*Lyapunov stability of ground states of nonlinear dispersive evolution equations*, Comm. Pure Appl. Math.**39**(1986), no. 1, 51–67. MR**820338**, DOI 10.1002/cpa.3160390103 - V. E. Zakharov and A. B. Shabat,
*Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media*, Ž. Èksper. Teoret. Fiz.**61**(1971), no. 1, 118–134 (Russian, with English summary); English transl., Soviet Physics JETP**34**(1972), no. 1, 62–69. MR**0406174**

**263**(2000), 131–156. KDV D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag.

**539**(1895), 422–443.

**79**(2000), 339–425. MM3 Y. Martel and F. Merle, Asymptotic stability of solitons for the subcritical generalized Korteweg–de Vries equation, to appear in Archive for Rational Mechanics and Analysis.

**316**(2000) 1, 103–137.

## Additional Information

**Frank Merle**- Affiliation: Département de Mathématiques, Université de Cergy–Pontoise, 2, avenue Adolphe Chauvin, BP 222, 95302 Cergy–Pontoise, France
- MR Author ID: 123710
- Received by editor(s): July 25, 2000
- Received by editor(s) in revised form: November 1, 2000
- Published electronically: March 20, 2001
- © Copyright 2001 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**14**(2001), 555-578 - MSC (2000): Primary 35B35, 35Q53
- DOI: https://doi.org/10.1090/S0894-0347-01-00369-1
- MathSciNet review: 1824989