{"title":"Stabilization in a chemotaxis system modelling T-cell dynamics with simultaneous production and consumption of signals","authors":"Youshan Tao, Michael Winkler","doi":"10.1017/s0956792524000299","DOIUrl":null,"url":null,"abstract":"In a smoothly bounded domain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline1.png\"/> <jats:tex-math> $\\Omega \\subset \\mathbb{R}^n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline2.png\"/> <jats:tex-math> $n\\ge 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system<jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0956792524000299_eqnU1.png\"/> <jats:tex-math> \\begin{eqnarray*} \\left \\{ \\begin{array}{l} u_t = \\Delta u - \\nabla \\cdot (u\\nabla v), \\\\[5pt] v_t = \\Delta v + u - \\alpha uv, \\end{array} \\right . \\end{eqnarray*} </jats:tex-math> </jats:alternatives> </jats:disp-formula>with parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline3.png\"/> <jats:tex-math> $\\alpha \\gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline4.png\"/> <jats:tex-math> $\\delta _\\star =\\delta _\\star (n)\\gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that for any given <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline5.png\"/> <jats:tex-math> $\\alpha \\ge \\frac{1}{\\delta _\\star }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and for any suitably regular initial data satisfying <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline6.png\"/> <jats:tex-math> $v(\\cdot, 0)\\le \\delta _\\star$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this problem admits a unique classical solution that stabilizes to the constant equilibrium <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792524000299_inline7.png\"/> <jats:tex-math> $(\\frac{1}{|\\Omega |}\\int _\\Omega u(\\cdot, 0), \\, \\frac{1}{\\alpha })$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in the large time limit.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0956792524000299","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
In a smoothly bounded domain $\Omega \subset \mathbb{R}^n$ , $n\ge 1$ , this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system \begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*} with parameter $\alpha \gt 0$ and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists $\delta _\star =\delta _\star (n)\gt 0$ such that for any given $\alpha \ge \frac{1}{\delta _\star }$ and for any suitably regular initial data satisfying $v(\cdot, 0)\le \delta _\star$ , this problem admits a unique classical solution that stabilizes to the constant equilibrium $(\frac{1}{|\Omega |}\int _\Omega u(\cdot, 0), \, \frac{1}{\alpha })$ in the large time limit.