{"title":"A critical blow-up exponent for flux limiation in a Keller-Segel system","authors":"M. Winkler","doi":"10.1512/iumj.2022.71.9042","DOIUrl":null,"url":null,"abstract":"The parabolic-elliptic cross-diffusion system \\[ \n\\left\\{ \\begin{array}{l} \nu_t = \\Delta u - \\nabla \\cdot \\Big(uf(|\\nabla v|^2) \\nabla v \\Big), \\\\[1mm] \n0 = \\Delta v - \\mu + u, \n\\qquad \\int_\\Omega v=0, \n\\qquad \n\\mu:=\\frac{1}{|\\Omega|} \\int_\\Omega u dx, \n\\end{array} \\right. \\] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $\\Omega\\subset R^n$, $n\\ge 1$, where $f$ generalizes the prototype given by \\[ \nf(\\xi) = (1+\\xi)^{-\\alpha}, \n\\qquad \\xi\\ge 0, \n\\qquad \\mbox{for all } \\xi\\ge 0, \\] with $\\alpha\\in R$. \nIn this framework, the main results assert that if $n\\ge 2$, $\\Omega$ is a ball and \\[ \n\\alpha<\\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. \nThis is complemented by a second statement which ensures that in general and not necessarily symmetric settings, if either $n=1$ and $\\alpha\\in R$ is arbitrary, or $n\\ge 2$ and $\\alpha>\\frac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2020-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"20","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indiana University Mathematics Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1512/iumj.2022.71.9042","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 20
Abstract
The parabolic-elliptic cross-diffusion system \[
\left\{ \begin{array}{l}
u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm]
0 = \Delta v - \mu + u,
\qquad \int_\Omega v=0,
\qquad
\mu:=\frac{1}{|\Omega|} \int_\Omega u dx,
\end{array} \right. \] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $\Omega\subset R^n$, $n\ge 1$, where $f$ generalizes the prototype given by \[
f(\xi) = (1+\xi)^{-\alpha},
\qquad \xi\ge 0,
\qquad \mbox{for all } \xi\ge 0, \] with $\alpha\in R$.
In this framework, the main results assert that if $n\ge 2$, $\Omega$ is a ball and \[
\alpha<\frac{n-2}{2(n-1)}, \] 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 $\alpha\in R$ is arbitrary, or $n\ge 2$ and $\alpha>\frac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.