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Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. (English) Zbl 1005.35081

The generalized KdV equation \(u_t+(u_{xx}+ u^p)_x=0\) \((0\leq t,-\infty<x <\infty\), \(p\geq 2\) is an integer), \(u(0,x)= u_0(x)\in H^1(\mathbb{R})\), has global and bounded in time solutions \(u(t)\in H^1(\mathbb{R})\) if \(p<5\). The article is devoted to the critical case \(p=5\).
Let \(Q=3^{1/4}/ \cosh^{1/2}2x\) be the critical state (solution of \(Q_{xx}+Q^5=Q)\) and \(E(u)=\frac 12\int u_x^2- \frac 16\int u^6\) be the energy. Assuming \(|u_0|_{L^2} <|Q|_{L^2}\), the solution \(u(t)\in H^1(\mathbb{R})\) is global and uniformly bounded. On the other hand, if \(|u_0|_{L^2} >|Q|_{L^2}\), then there exists \(\alpha_0 >0\) such that for all \(u_0\in H^1(\mathbb{R})\) with \(E(u_0)<0\) and \(\int u_0^2\leq\int Q^2+ \alpha_0\), there are blow-up solutions \(|u(t)|_{H^1} \to\infty\) as \(t\to T\) (where \(T\leq\infty)\). In the latter case, the article involves two qualitative results.
Theorem on the stability of the blow-up profile: There exist \(\lambda(t)\geq 0\) and \(x(t)\in\mathbb{R}\) such that the function \(\lambda^{ 1/2} (t)u(t,\lambda (t)x-x(t))\) weakly converges in \(H^1(\mathbb{R})\) either to \(Q\) or to \(-Q\) as \(t\to T\).
Theorem on the blow-up rate: \(\lim(T-t)^{1/3}|u_x(t) |_{L^2}= \infty\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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