论文标题
Keller-Segel系统中通量限制的关键爆炸指数
A critical blow-up exponent for flux limitation in a Keller-Segel system
论文作者
论文摘要
抛物线 - 涡旋交叉扩散系统\ [ \ left \ {\ begin {array} {l} u_t =ΔU-\ nabla \ cdot \ big（uf（| \ nabla v |^2）\ nabla v \ big），\\ [1mm] 0 =ΔV -μ+ u， \ qquad \int_Ωv= 0， \ Qquad μ：= \ frac {1} {|ω|} \int_Ωudx， \ end {array} \ right。 \]在平滑界面$ω\ subset r^n $，$ n \ ge 1 $中的均匀的neumann-type边界条件一起考虑，其中$ f $概述了\ [ f（ξ）=（1+ξ）^{ - α}， \qquadξ\ ge 0， \ qquad \ mbox {对于所有}ξ\ ge 0，\ \]，$α\在r $中。 在此框架中，主要结果断言，如果$ n \ ge 2 $，$ω$是一个球，\ [ 然后在一组相当大的径向对称初始数据中，α<\ frac {n-2} {2（n-1）}，\ \ \ \]，相关的初始值问题允许相对于其第一个组件的$ l^\ infty $ norm在有限的时间内爆炸的解决方案。 这是通过第二个陈述来补充的，该声明确保一般而言，不一定是对称的设置，如果$ n = 1 $和$α\ in r $中是任意的，或$ n \ ge 2 $和$α> \ frac {n-2} {n-2} {2（n-1）} $，则在义务上统治了任何企业，因此，任何爆炸都在不连续的情况下进行了裁定，并连续地进行了仲裁，并且是不断的，并且是连续的。
The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = Δu - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = Δv - μ+ u, \qquad \int_Ωv=0, \qquad μ:=\frac{1}{|Ω|} \int_Ωu dx, \end{array} \right. \] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $Ω\subset R^n$, $n\ge 1$, where $f$ generalizes the prototype given by \[ f(ξ) = (1+ξ)^{-α}, \qquad ξ\ge 0, \qquad \mbox{for all } ξ\ge 0, \] with $α\in R$. In this framework, the main results assert that if $n\ge 2$, $Ω$ is a ball and \[ α<\frac{n-2}{2(n-1)}, \] then throughout a considerably large set of radially symmetric initial data, an associated initial value problem admits solutions blowing up in finite time with respect to the $L^\infty$ norm of their first components. This is complemented by a second statement which ensures that in general and not necessarily symmetric settings, if either $n=1$ and $α\in R$ is arbitrary, or $n\ge 2$ and $α>\frac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.
