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

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

Yulan Wang, Michael Winkler

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<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":"{\"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}","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}

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

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

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