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<p>In this work, we investigate a chemotaxis system that describes the interplay between attraction and repulsion mechanisms. The model involves two species and two chemical substances under Neumann boundary conditions, incorporating a singular sensitivity in the parabolic response function. These chemicals act as signaling agents: they attract the species at high concentrations and repel them at low concentrations, and both signals are produced by the same species. Based on the proposed model, we establish the existence of a global classical solution by applying standard semigroup estimates for parabolic equations, provided that <Equation ID="Equ114"> <EquationSource Format="TEX">\(\begin{aligned} {\left\{ \begin{array}{ll} { 0&lt; }~\beta _{1},\beta _{2}&lt;2 &amp; \text { if } \ N=1\\ { 0&lt; }~ \beta _{1},\beta _{2}&lt; 1- \frac{N}{4} &amp; \text { if } \ N\ge 2. \end{array}\right. } \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mrow> <mn>0</mn> <mo>&lt;</mo> </mrow> <mspace width="3.33333pt" /> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>2</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>if</mtext> <mspace width="0.333333em" /> <mspace width="4pt" /> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mrow> <mn>0</mn> <mo>&lt;</mo> </mrow> <mspace width="3.33333pt" /> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>1</mn> <mo>-</mo> <mfrac> <mi>N</mi> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>if</mtext> <mspace width="0.333333em" /> <mspace width="4pt" /> <mi>N</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation></p>

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Global solvability of a two-species attraction-repulsion chemotaxis system with singular sensitivity

  • 期刊论文

  • 发表日期：

    2026年2月27日

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  • S. Amalorpava Josephine,
  • S. Karthikeyan,
  • L. Shangerganesh,
  • K. Yadhavan

摘要

In this work, we investigate a chemotaxis system that describes the interplay between attraction and repulsion mechanisms. The model involves two species and two chemical substances under Neumann boundary conditions, incorporating a singular sensitivity in the parabolic response function. These chemicals act as signaling agents: they attract the species at high concentrations and repel them at low concentrations, and both signals are produced by the same species. Based on the proposed model, we establish the existence of a global classical solution by applying standard semigroup estimates for parabolic equations, provided that \(\begin{aligned} {\left\{ \begin{array}{ll} { 0< }~\beta _{1},\beta _{2}<2 & \text { if } \ N=1\\ { 0< }~ \beta _{1},\beta _{2}< 1- \frac{N}{4} & \text { if } \ N\ge 2. \end{array}\right. } \end{aligned}\) 0 < β 1 , β 2 < 2 if N = 1 0 < β 1 , β 2 < 1 - N 4 if N ≥ 2 .