{"title":"Complete infinite-time mass aggregation in a quasilinear Keller–Segel system","authors":"Michael Winkler","doi":"10.1007/s11856-024-2618-9","DOIUrl":null,"url":null,"abstract":"<p>Radially symmetric global unbounded solutions of the chemotaxis system </p><span>$$\\left\\{ {\\matrix{{{u_t} = \\nabla \\cdot (D(u)\\nabla u) - \\nabla \\cdot (uS(u)\\nabla v),} \\hfill & {} \\hfill \\cr {0 = \\Delta v - \\mu + u,} \\hfill & {\\mu = {1 \\over {|\\Omega |}}\\int_\\Omega {u,} } \\hfill \\cr } } \\right.$$</span><p> are considered in a ball Ω = <i>B</i><sub><i>R</i></sub>(0) ⊂ ℝ<sup><i>n</i></sup>, where <i>n</i> ≥ 3 and <i>R</i> > 0.</p><p>Under the assumption that <i>D</i> and <i>S</i> suitably generalize the prototypes given by <i>D</i>(<i>ξ</i>) = (<i>ξ</i> + <i>ι</i>)<sup>m−1</sup> and <i>S</i>(<i>ξ</i>) = (<i>ξ</i> + 1)<sup>−λ−1</sup> for all <i>ξ</i> > 0 and some <i>m</i> ∈ ℝ, λ >0 and <i>ι</i> ≥ 0 fulfilling <span>\\(m + \\lambda < 1 - {2 \\over n}\\)</span>, a considerably large set of initial data <i>u</i><sub>0</sub> is found to enforce a complete mass aggregation in infinite time in the sense that for any such <i>u</i><sub>0</sub>, an associated Neumann type initial-boundary value problem admits a global classical solution (<i>u, v</i>) satisfying </p><span>$${1 \\over C} \\cdot {(t + 1)^{{1 \\over \\lambda }}} \\le ||u( \\cdot ,t)|{|_{{L^\\infty }(\\Omega )}} \\le C \\cdot {(t + 1)^{{1 \\over \\lambda }}}\\,\\,\\,{\\rm{for}}\\,\\,{\\rm{all}}\\,\\,t > 0$$</span><p> as well as </p><span>$$||u( \\cdot \\,,t)|{|_{{L^1}(\\Omega \\backslash {B_{{r_0}}}(0))}} \\to 0\\,\\,\\,{\\rm{as}}\\,\\,t \\to \\infty \\,\\,\\,{\\rm{for}}\\,\\,{\\rm{all}}\\,\\,{r_0} \\in (0,R)$$</span><p> with some <i>C</i> > 0.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2618-9","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Radially symmetric global unbounded solutions of the chemotaxis system
are considered in a ball Ω = BR(0) ⊂ ℝn, where n ≥ 3 and R > 0.
Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)m−1 and S(ξ) = (ξ + 1)−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling \(m + \lambda < 1 - {2 \over n}\), a considerably large set of initial data u0 is found to enforce a complete mass aggregation in infinite time in the sense that for any such u0, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying
期刊介绍:
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.