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

By F. Merle. 2001

171 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2d08b31f2defd042d118ae48633a21720bc9fa79

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Multi-Soliton Solutions for the Supercritical gKdV Equations (2010, background): “[14] Y. Martel and F. Merle.”
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case (2006, background): “(See [19] for similar proof) Assume that u0 6≡ 0 so that ∫ |▽u0| 2 > 0 and because of Lemma 5.”
On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping (2007, background): “On the other hand, in [10] Merle showed that H(1) solutions may blow-up in finite time.”
Blow up for the critical gKdV equation III: exotic regimes (2012, background): “It is proved in [44, 33] that initial data u0 such that (1.”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, methodology): “This observation was used in [73, Theorem 3.3] and also in [35, 40 ].”
On the Cauchy Problem for the Generalized Korteweg-de Vries Equation (2003, background): “On the other hand it has been recently proved in Martel and Merle ( [2002] ), Merle ( [2001] ) that blow up in finite time can occur in the critical case k = 4.”
Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions (2007, methodology): “3 were obtained for the energy-critical NLS in [17] and for the mass-critical generalized Korteweg–de Vries equation in [19].”
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation (2005, background): “It has been proved in the papers [13], [14], [15], [16], [17] that for α0 small enough, if E(u0) < 0 and ∫ |u0| < ∫ Q2 + α0, then: (i) Blow-up occurs in finite or infinite time, i.”
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation (2006, background): “1 of [16] (also see [22] for a similar proof).”
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation (2003, background): “in the sequel of papers [17], [18], [19], [20], [21].”
Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gKdV Equations (2001, background): “This is in contrast with the case p = 5, for which there exist solutions u(t) of (1) such that |u(t)|H1 → +∞ as t→ T , for 0 < T < +∞, see [19] and [17].”