{"title":"具有信号依赖性退化扩散和逻辑源的间接趋化-消费模型的全局经典解法","authors":"Meng Zheng, Liangchen Wang","doi":"10.1007/s00033-024-02303-x","DOIUrl":null,"url":null,"abstract":"<p>This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{llll} u_t = \\Delta \\left( u v^\\alpha \\right) +au-bu^l,\\quad &{}x\\in \\Omega ,t>0,\\\\ v_t= \\Delta v - vw,\\quad &{}x\\in \\Omega ,t>0,\\\\ w_t = - \\delta w + u,\\quad &{}x\\in \\Omega ,t>0, \\end{array} \\right. \\end{aligned}$$</span><p>under homogeneous Neumann boundary conditions in a smooth bounded domain <span>\\(\\Omega \\subset \\mathbb {R}^n\\)</span> (<span>\\(n\\ge 1\\)</span>). Here, the parameters <span>\\(a>0\\)</span>, <span>\\(b>0\\)</span>, <span>\\(\\alpha \\ge 1\\)</span>, <span>\\(\\delta >0\\)</span> and <span>\\(l \\ge 2\\)</span>. For all suitably regular initial data, if one of the following cases holds: </p><ol>\n<li>\n<span>(i)</span>\n<p><span>\\(l > 2\\)</span>;</p>\n</li>\n<li>\n<span>(ii)</span>\n<p><span>\\(l =2, n\\le 3\\)</span>;</p>\n</li>\n<li>\n<span>(iii)</span>\n<p><span>\\(l = 2, n \\ge 4,\\)</span> and <i>b</i> is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution.</p>\n</li>\n</ol>","PeriodicalId":501481,"journal":{"name":"Zeitschrift für angewandte Mathematik und Physik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Global classical solutions to an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source\",\"authors\":\"Meng Zheng, Liangchen Wang\",\"doi\":\"10.1007/s00033-024-02303-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source </p><span>$$\\\\begin{aligned} \\\\left\\\\{ \\\\begin{array}{llll} u_t = \\\\Delta \\\\left( u v^\\\\alpha \\\\right) +au-bu^l,\\\\quad &{}x\\\\in \\\\Omega ,t>0,\\\\\\\\ v_t= \\\\Delta v - vw,\\\\quad &{}x\\\\in \\\\Omega ,t>0,\\\\\\\\ w_t = - \\\\delta w + u,\\\\quad &{}x\\\\in \\\\Omega ,t>0, \\\\end{array} \\\\right. \\\\end{aligned}$$</span><p>under homogeneous Neumann boundary conditions in a smooth bounded domain <span>\\\\(\\\\Omega \\\\subset \\\\mathbb {R}^n\\\\)</span> (<span>\\\\(n\\\\ge 1\\\\)</span>). Here, the parameters <span>\\\\(a>0\\\\)</span>, <span>\\\\(b>0\\\\)</span>, <span>\\\\(\\\\alpha \\\\ge 1\\\\)</span>, <span>\\\\(\\\\delta >0\\\\)</span> and <span>\\\\(l \\\\ge 2\\\\)</span>. For all suitably regular initial data, if one of the following cases holds: </p><ol>\\n<li>\\n<span>(i)</span>\\n<p><span>\\\\(l > 2\\\\)</span>;</p>\\n</li>\\n<li>\\n<span>(ii)</span>\\n<p><span>\\\\(l =2, n\\\\le 3\\\\)</span>;</p>\\n</li>\\n<li>\\n<span>(iii)</span>\\n<p><span>\\\\(l = 2, n \\\\ge 4,\\\\)</span> and <i>b</i> is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution.</p>\\n</li>\\n</ol>\",\"PeriodicalId\":501481,\"journal\":{\"name\":\"Zeitschrift für angewandte Mathematik und Physik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Zeitschrift für angewandte Mathematik und Physik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00033-024-02303-x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Zeitschrift für angewandte Mathematik und Physik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00033-024-02303-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
本文讨论了以下间接趋化-消耗模型,该模型具有信号依赖性退化扩散和对数源 $$\begin{aligned}(开始{array}{llll})。\u_t = \Delta \left( u v^\alpha \right) +au-bu^l,\quad &{}x\in \Omega ,t>;0,\v_t= \Delta v - vw,\quad &{}x\in \Omega ,t>0,\w_t = - \delta w + u,\quad &{}x\in \Omega ,t>0, (end{array}.\right.\end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\) (\(n\ge 1\)).这里,参数有\(a>0\),\(b>0\),\(α\ge 1\),\(\delta >0\)和\(l\ge 2\).对于所有适当规则的初始数据,如果以下情况之一成立:(i)\(l > 2\);(ii)\(l =2, n\le 3\);(iii)\(l = 2, n\ge 4,\)并且b足够大,那么相应的初始边界值问题就拥有一个全局经典解。
under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\) (\(n\ge 1\)). Here, the parameters \(a>0\), \(b>0\), \(\alpha \ge 1\), \(\delta >0\) and \(l \ge 2\). For all suitably regular initial data, if one of the following cases holds:
(i)
\(l > 2\);
(ii)
\(l =2, n\le 3\);
(iii)
\(l = 2, n \ge 4,\) and b is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution.