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

Keywords:

KdV equation; blow-up solutions
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