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<p>In the past decade, there has been much interest in analyzing Keller-Segel models with tensorial flux. However, it is yet not well understood whether there are solutions that blow-up in a finite amount of time. We aim to prove the possibility of having solutions blowing up in finite time when having a tensorial flux of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1019_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(A\nabla v\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi mathvariant="normal">∇</mi> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>, where <i>A</i> represents an arbitrary matrix with constant components satisfying <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1019_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </InlineMediaObject> <EquationSource Format="TEX">\(Tr(A),\det (A)&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo movablelimits="true">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A note on the blow-up of solutions to the two-dimensional Keller-Segel model with tensorial flux

  • 期刊论文

  • 发表日期：

    2025年1月27日

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  • Valeria Cuentas,
  • Elio Espejo

摘要

In the past decade, there has been much interest in analyzing Keller-Segel models with tensorial flux. However, it is yet not well understood whether there are solutions that blow-up in a finite amount of time. We aim to prove the possibility of having solutions blowing up in finite time when having a tensorial flux of the form \(A\nabla v\) A ∇ v , where A represents an arbitrary matrix with constant components satisfying \(Tr(A),\det (A)>0\) T r ( A ) , det ( A ) > 0 .