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

Abstract

In this paper, we consider a blow-up solution $u(t)$ (close to the soliton manifold) to the $L^2$-critical gKdV equation ${\partial }_{t}u+(u_{xx}+u^5{)}_x=0$, 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 ${\partial }_{t}u+(u_{xx}+u^5-\gamma u|u{|}^{q-1}{)}_x=0$ 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 $\gamma \to 0$. We prove in this paper that for all $t\ge T$, $u_{\gamma }(t)\to v(t)$ as $\gamma \to 0$, 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.