Existence of blow-up solutions in the energy space for the critical generalized KdV equation (Q2719031)

Generalized Korteweg-de Vries equations (KdV) appear in the study of waves on shallow water [\textit{D.J.Korteweg and \textit{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: NEWLINE\[NEWLINEu_t +(u_{xx} + u^5)_{x} =0,\quad (t,x)\in \mathbb{R}^+\times\mathbb{R}.NEWLINE\]NEWLINE 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) NEWLINE\[NEWLINEiu_t = - u_{xx} -|u|^4u,\quad (t,x)\in\mathbb{R}^+\times \mathbb{R}.NEWLINE\]NEWLINE Both equations satisfy two identical conservation laws. The NLSE was shown by \textit{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. NEWLINENEWLINENEWLINEA 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 [\textit{Y. Martel} and \textit{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.NEWLINENEWLINENEWLINEThere exists \(\alpha_0>0\) such that, if \(u_0 \in H^1(\mathbb{R})\) and the solution \(u(t)\) of the critical KdV satisfy: NEWLINE\[NEWLINEE(u_0)<0\qquad\text{and}\qquad \int u_0^2 < \int Q^2+\alpha_0,NEWLINE\]NEWLINE 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 \(E(v)={1\over 2}\int v_x^2 -{1\over 6}\int v^6\).}