当前位置: X-MOL 学术Rev. Mat. Iberoam. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On continuation properties after blow-up time for $L^2$-critical gKdV equations
Revista Matemática Iberoamericana ( IF 1.3 ) Pub Date : 2020-01-07 , DOI: 10.4171/rmi/1154
Yang Lan 1
Affiliation  

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:

$ 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 $。
更新日期:2020-01-07
down
wechat
bug