A singular growth phenomenon in a Keller–Segel–type parabolic system involving density-suppressed motilities

Pub Date : 2024-02-23 DOI:10.1002/mana.202300361

Yulan Wang, Michael Winkler

{"title":"A singular growth phenomenon in a Keller–Segel–type parabolic system involving density-suppressed motilities","authors":"Yulan Wang, Michael Winkler","doi":"10.1002/mana.202300361","DOIUrl":null,"url":null,"abstract":"A no-flux initial-boundary value problem for\n\n Under the assumption that <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>></mo>\n <mfrac>\n <mi>n</mi>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation>$\\alpha &gt;\\frac{n}{n-2}$</annotation>\n </semantics></math>, it is shown that for each <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$m&gt;0$</annotation>\n </semantics></math>, there exist <math>\n <semantics>\n <mrow>\n <mi>T</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$T&gt;0$</annotation>\n </semantics></math> and a positive <math>\n <semantics>\n <mrow>\n <msub>\n <mi>v</mi>\n <mn>0</mn>\n </msub>\n <mo>∈</mo>\n <msup>\n <mi>W</mi>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mi>∞</mi>\n </mrow>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>Ω</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$v_0\\in W^{1,\\infty }(\\Omega)$</annotation>\n </semantics></math> with the property that whenever <math>\n <semantics>\n <mrow>\n <msub>\n <mi>u</mi>\n <mn>0</mn>\n </msub>\n <mo>∈</mo>\n <msup>\n <mi>W</mi>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mi>∞</mi>\n </mrow>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>Ω</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$u_0\\in W^{1,\\infty }(\\Omega)$</annotation>\n </semantics></math> is nonnegative with <math>\n <semantics>\n <mrow>\n <msub>\n <mo>∫</mo>\n <mi>Ω</mi>\n </msub>\n <msub>\n <mi>u</mi>\n <mn>0</mn>\n </msub>\n <mo>=</mo>\n <mi>m</mi>\n </mrow>\n <annotation>$\\int _\\Omega u_0=m$</annotation>\n </semantics></math>, the global solutions to (<math>\n <semantics>\n <mi>★</mi>\n <annotation>$\\star$</annotation>\n </semantics></math>) emanating from the initial data <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>u</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>v</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(u_0,v_0)$</annotation>\n </semantics></math> have the property that\n\n ","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300361","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300361","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

A no-flux initial-boundary value problem for

Under the assumption that $α > \frac{n}{n - 2}$ , it is shown that for each $m > 0$ , there exist $T > 0$ and a positive $v_{0} \in W^{1, \infty} (Ω)$ with the property that whenever $u_{0} \in W^{1, \infty} (Ω)$ is nonnegative with $\int_{Ω} u_{0} = m$ , the global solutions to ( $★$ ) emanating from the initial data $(u_{0}, v_{0})$ have the property that

查看原文

涉及密度抑制运动的凯勒-西格尔型抛物线系统中的奇异增长现象

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文