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{"title":"模拟非线性扩散溶瘤病毒治疗的触致交叉扩散系统的全局弱解","authors":"Yue Zhou, Changchun Liu","doi":"10.1007/s00245-025-10232-y","DOIUrl":null,"url":null,"abstract":"<div><p>This paper discusses an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy </p><div><div><span>$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t=\\Delta u^m-\\nabla \\cdot (u\\nabla v) +\\mu u(1-u)-uz,\\;\\;& \\;x\\in \\Omega ,~t>0, \\\\ v_t=-(u+w)v,\\;\\;& \\;x\\in \\Omega ,~t>0, \\\\ w_t=\\Delta w-\\nabla \\cdot (w\\nabla v)-w+uz,\\;\\;& \\;x\\in \\Omega ,~t>0, \\\\ z_t=D\\Delta z-z-uz+\\beta w,\\;\\;& \\;x\\in \\Omega ,~t>0, \\\\ \\frac{\\partial u^m}{\\partial \\nu }-u\\frac{\\partial v}{\\partial \\nu }=\\frac{\\partial w}{\\partial \\nu }-w\\frac{\\partial v}{\\partial \\nu }=\\frac{\\partial z}{\\partial \\nu }=0,\\;\\;& \\;x\\in \\partial \\Omega ,~t>0, \\\\ u(x,0)=u_{0},~v(x,0)=v_{0},~w(x,0)=w_{0}, \\\\ z(x,0)=z_{0},\\;\\;& \\;x\\in \\Omega , \\end{array}\\right. } \\end{aligned}$$</span></div></div><p>in a smooth bounded domain <span>\\( \\Omega \\subset {\\mathbb {R}}^{N}(N=1,2) \\)</span> with <span>\\( m>1, \\beta>0, \\mu >0 \\)</span>, and <span>\\( D>0\\)</span>. We prove that for any large initial datum, the problem admits a global ‘very’ weak solution for any <span>\\(m>1\\)</span>.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Global Weak Solutions in a Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy with Nonlinear Diffusion\",\"authors\":\"Yue Zhou, Changchun Liu\",\"doi\":\"10.1007/s00245-025-10232-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper discusses an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy </p><div><div><span>$$\\\\begin{aligned} {\\\\left\\\\{ \\\\begin{array}{ll} u_t=\\\\Delta u^m-\\\\nabla \\\\cdot (u\\\\nabla v) +\\\\mu u(1-u)-uz,\\\\;\\\\;& \\\\;x\\\\in \\\\Omega ,~t>0, \\\\\\\\ v_t=-(u+w)v,\\\\;\\\\;& \\\\;x\\\\in \\\\Omega ,~t>0, \\\\\\\\ w_t=\\\\Delta w-\\\\nabla \\\\cdot (w\\\\nabla v)-w+uz,\\\\;\\\\;& \\\\;x\\\\in \\\\Omega ,~t>0, \\\\\\\\ z_t=D\\\\Delta z-z-uz+\\\\beta w,\\\\;\\\\;& \\\\;x\\\\in \\\\Omega ,~t>0, \\\\\\\\ \\\\frac{\\\\partial u^m}{\\\\partial \\\\nu }-u\\\\frac{\\\\partial v}{\\\\partial \\\\nu }=\\\\frac{\\\\partial w}{\\\\partial \\\\nu }-w\\\\frac{\\\\partial v}{\\\\partial \\\\nu }=\\\\frac{\\\\partial z}{\\\\partial \\\\nu }=0,\\\\;\\\\;& \\\\;x\\\\in \\\\partial \\\\Omega ,~t>0, \\\\\\\\ u(x,0)=u_{0},~v(x,0)=v_{0},~w(x,0)=w_{0}, \\\\\\\\ z(x,0)=z_{0},\\\\;\\\\;& \\\\;x\\\\in \\\\Omega , \\\\end{array}\\\\right. } \\\\end{aligned}$$</span></div></div><p>in a smooth bounded domain <span>\\\\( \\\\Omega \\\\subset {\\\\mathbb {R}}^{N}(N=1,2) \\\\)</span> with <span>\\\\( m>1, \\\\beta>0, \\\\mu >0 \\\\)</span>, and <span>\\\\( D>0\\\\)</span>. We prove that for any large initial datum, the problem admits a global ‘very’ weak solution for any <span>\\\\(m>1\\\\)</span>.</p></div>\",\"PeriodicalId\":55566,\"journal\":{\"name\":\"Applied Mathematics and Optimization\",\"volume\":\"91 2\",\"pages\":\"\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2025-02-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics and Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00245-025-10232-y\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-025-10232-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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摘要
讨论了由溶瘤病毒治疗引起的双触致交叉扩散系统的初边值问题 $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u^m-\nabla \cdot (u\nabla v) +\mu u(1-u)-uz,\;\;& \;x\in \Omega ,~t>0, \\ v_t=-(u+w)v,\;\;& \;x\in \Omega ,~t>0, \\ w_t=\Delta w-\nabla \cdot (w\nabla v)-w+uz,\;\;& \;x\in \Omega ,~t>0, \\ z_t=D\Delta z-z-uz+\beta w,\;\;& \;x\in \Omega ,~t>0, \\ \frac{\partial u^m}{\partial \nu }-u\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }-w\frac{\partial v}{\partial \nu }=\frac{\partial z}{\partial \nu }=0,\;\;& \;x\in \partial \Omega ,~t>0, \\ u(x,0)=u_{0},~v(x,0)=v_{0},~w(x,0)=w_{0}, \\ z(x,0)=z_{0},\;\;& \;x\in \Omega , \end{array}\right. } \end{aligned}$$在光滑有界区域中 \( \Omega \subset {\mathbb {R}}^{N}(N=1,2) \) 有 \( m>1, \beta>0, \mu >0 \),和 \( D>0\). 我们证明,对于任何大的初始数据,问题承认一个全局“非常”弱解 \(m>1\).
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