Revista Matemática Iberoamericana


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Published online first: 2020-01-07
DOI: 10.4171/rmi/1154

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

Yang Lan[1]

(1) Universität Basel, Switzerland

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.

Keywords: gKdV, $L^2$-critical, blow-up, continuation after blow-up

Lan Yang: On continuation properties after blow-up time for $L^2$-critical gKdV equations. Rev. Mat. Iberoam. Electronically published on January 7, 2020. doi: 10.4171/rmi/1154 (to appear in print)