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arXiv 1709.09535math.AP
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关于$L^2$ -临界gKdV方程爆破后的延拓性质

On continuation properties after blow-up time for $L^2$-critical gKdV equations

Yang Lan

备注：
24 pages, minor revision
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摘要：

本文考虑$L^2$ -临界gKdV方程$\partial_tu+(u_{xx}+u^5)_x=0$的爆破解$u(t)$，爆破时间为有限$T<+\infty$ .我们期望在blow-up时间之后构造$u(t)$的自然扩展。为此，我们考虑具有相同初始数据的饱和$L^2$ -临界gKdV方程$\partial_tu+(u_{xx}+u^5-γu|u|^{q-1})_x=0$的解$u_γ(t)$，其中$γ>0$和$q>5$。一个标准参数表明$u_γ(t)$在时间上总是全局的，并且对于所有$t<T$，$u_γ(t)$收敛到$H^1$中的$u(t)$，作为$γ\rightarrow0$。本文证明了对所有的$t\geq T$，$u_γ(t)$在一定意义下收敛到某个$v(t)$为$γ\rightarrow0$.这个极限函数$v(t)$是未扰动的$L^2$临界gKdV的弱解，因此可以看作是爆破时间后$u(t)$的自然扩展。

英文摘要：

In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T<+\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_γ(t)$ to the saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-γu|u|^{q-1})_x=0$ with the same initial data, where $γ>0$ and $q>5$. A standard argument shows that $u_γ(t)$ is always global in time and for all $t<T$, $u_γ(t)$ converges to $u(t)$ in $H^1$ as $γ\rightarrow0$. We prove in this paper that for all $t\geq T$, $u_γ(t)$ converges to some $v(t)$ as $γ\rightarrow0$, in a certain sense. This limiting function $v(t)$ is a weak solution to the unperturbed $L^2$-critical gKdV, hence can be viewed as a natural extension of $u(t)$ after the blow-up time.

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