临界广义KdV方程的慢行进无穷点爆破
Slowly travelling infinite point blow-up for the critical generalized KdV equation
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中文总结 AI 辅助
本文研究临界广义KdV方程的有限时间爆破，证明存在慢行进无穷点爆破解，速率阈值ν=1/2，采用指数尾项并调整能量-位力泛函，扩展了相关爆破解的构造结果。
中文摘要 AI 辅助
我们研究五次质量临界广义KdV（gKdV）方程的有限时间爆破现象。我们证明存在一类解U，其具有无穷点、有限时间爆破行为，爆破速率满足：当t趋近于T时，‖∂ₓU(t)‖_{L²} ~ (T-t)^{-ν}，其中ν=1/2，T为爆破时间，且爆破泡的行进速度呈对数形式。因此，我们将此行为称为慢行进无穷点爆破。特殊爆破速率ν=1/2是区分有限与无穷点泡化的阈值。在之前的工作arXiv:2511.13538中，作者针对连续区间ν∈(1/2,1)构造了其他无穷点爆破解，利用空间变量的多项式尾项并扩展了arXiv:1209.2510的结果（该结果限于ν>11/13）。然而，该工作指出阈值情形下尾项需发生变化。在本文中，我们考虑空间右侧的指数衰减尾项。与arXiv:2511.13538类似，初始数据可取自H¹中任意接近基态的点。从技术角度看，除了尾项的改变，我们还需针对指数尾项调整能量-位力泛函，通过修改用于控制爆破解右侧的标度项实现。
英文摘要
We study the finite time blow up phenomenon for the quintic, mass critical gKdV equation. We prove the existence of a class of solutions $U$ with infinite point, finite time blow up behavior, at the particular blow up rate $$\|\partial_x U(t)\|_{L^2} \sim (T-t)^{-ν}\quad \text{as} \quad t \uparrow T,$$ where $ν= \frac 12$, $T$ is the blow up time and where the travel speed of the blow up bubble is logarithmic. Therefore, we call this behaviour slowly travelling infinite point blow up. The special blow up rate $ν=\frac 12$ is a threshold which separates finite and infinite point bubbling. In a previous work arXiv:2511.13538, the author constructed other infinite point blow up solutions for the continuum of rates $ν\in(\frac 12,1)$, using polynomial tails in the space variable and extending the results in arXiv:1209.2510, restricted to $ν> \frac{11}{13}$. However, that work suggested a change of the tail for the threshold case. In the present paper, we consider an exponentially decaying tail on the right in space. As in arXiv:2511.13538, the initial data can be taken arbitrarily close to the ground state in $H^1$. From a technical perspective, in addition to the change of tail, we have to adapt the energy-virial functional to the presence of the exponential tail, by modifying a scaling term used to control the right-hand side of the blow up solution.
