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

中文翻译：

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$ L ^ 2 $-临界gKdV方程在爆破时间后的连续性

在本文中，我们考虑了到$ L ^ 2 $临界gKdV方程$ \ partial_tu +（u_ {xx} + u ^ 5）_x =的爆破解$ u（t）$（靠近孤子流形）。 0 $，且爆破时间为$ T <+ \ infty $。我们期望在爆炸时间之后自然扩展$ u（t）$。为此，我们考虑对饱和$ L ^ 2 $临界gKdV方程$ \ partial_tu +（u_ {xx} + u ^ 5- \ gamma u | u | ^的解$ u _ {\ gamma}（t）$ {q-1}）_ x = 0 $具有相同的初始数据，其中$ \ gamma> 0 $和$ q> 5 $。一个标准的论证表明，$ u _ {\ gamma}（t）$在时间上始终是全局的。此外，对于所有$ t <T $，$ u _ {\ gamma}（t）$在$ H ^ 1 $中收敛为$ u（t）$作为$ \ gamma \ rightarrow0 $。我们在本文中证明，在一定意义上，对于所有$ t \ geq T $，$ u _ {\ gamma}（t）\ rightarrow v（t）$为$ \ gamma \ rightarrow0 $。