Existence of blow-up solutions in the energy space for the critical generalized KdV equation. (English) Zbl 0970.35128
Generalized Korteweg-de Vries equations (KdV) appear in the study of waves on shallow water [D.J.Korteweg and G.de Vries, Philos. Mag. (5) 39, 422-433 (1895; JFM 26.0881.02)]. In the present paper, the author is concerned with the existence of blow-up solutions of the so-called critical KdV:
\[
u_t +(u_{xx} + u^5)_{x} =0,\quad (t,x)\in \mathbb{R}^+\times\mathbb{R}.
\]
The blow-up of the solutions of the critical KdV was conjectured long ago because of their similarity with the critical nonlinear Schrödinger equation (NLSE)
\[
iu_t = - u_{xx} -|u|^4u,\quad (t,x)\in\mathbb{R}^+\times \mathbb{R}.
\]
Both equations satisfy two identical conservation laws. The NLSE was shown by M. I. Weinstein [SIAM J. Math. Anal. 16, 472-491, (1985; Zbl 0583.35028)], using the conformal structure of the NLSE, to possess an explicit solution that blows up.
A first step in the direction of proving the existence of blow-up solutions of the critical KdV was done recently by Martel and the author. They established the instability of some particular solutions of the critical KdV (solitons) in [Instability of solitons of the critical Kortweg-de Vries equation, to appear in Geom. Funct. Anal.]. These are known to be stable in the subcritical case and instable in the supercritical case. In [Y. Martel and F. Merle, J. Math. Pures Appl., IX. Sér. 79, No. 4, 339-425 (2000; Zbl 0963.37058)], Martel and the author analyzed the role of the dispersion and established that any global solution in \(H^1\), which is at the initial time close to a soliton, is precisely the soliton. The main result of this paper which is a continuation is the following.
There exists \(\alpha_0>0\) such that, if \(u_0 \in H^1(\mathbb{R})\) and the solution \(u(t)\) of the critical KdV satisfy: \[ E(u_0)<0\qquad\text{and}\qquad \int u_0^2 < \int Q^2+\alpha_0, \] then \(u(t)\) blows up in \(H^1\) in finite or infinite time. \(Q(x)={3^{1/4}\over ch ^{1/2}(2x)}\) (the ground state) is the solution of \(Q_{xx} + Q^5 =Q\) and \(E(v)={1\over 2}\int v_x^2 -{1\over 6}\int v^6\).
A first step in the direction of proving the existence of blow-up solutions of the critical KdV was done recently by Martel and the author. They established the instability of some particular solutions of the critical KdV (solitons) in [Instability of solitons of the critical Kortweg-de Vries equation, to appear in Geom. Funct. Anal.]. These are known to be stable in the subcritical case and instable in the supercritical case. In [Y. Martel and F. Merle, J. Math. Pures Appl., IX. Sér. 79, No. 4, 339-425 (2000; Zbl 0963.37058)], Martel and the author analyzed the role of the dispersion and established that any global solution in \(H^1\), which is at the initial time close to a soliton, is precisely the soliton. The main result of this paper which is a continuation is the following.
There exists \(\alpha_0>0\) such that, if \(u_0 \in H^1(\mathbb{R})\) and the solution \(u(t)\) of the critical KdV satisfy: \[ E(u_0)<0\qquad\text{and}\qquad \int u_0^2 < \int Q^2+\alpha_0, \] then \(u(t)\) blows up in \(H^1\) in finite or infinite time. \(Q(x)={3^{1/4}\over ch ^{1/2}(2x)}\) (the ground state) is the solution of \(Q_{xx} + Q^5 =Q\) and \(E(v)={1\over 2}\int v_x^2 -{1\over 6}\int v^6\).
Reviewer: Youssef Jabri (Oujda)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B35 | Stability in context of PDEs |
| 37K45 | Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
critical generalized KdV equation; existence of blow-up solutions; critical nonlinear Schrödinger equationReferences:
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