{"title":"模拟溶瘤病毒治疗的三维触觉交叉扩散系统的渐近行为","authors":"Yifu Wang, Chi Xu","doi":"10.1142/s0218202523400043","DOIUrl":null,"url":null,"abstract":"This paper deals with an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy \\begin{equation*} \\left\\{ \\begin{array}{lll} u_t=\\Delta u-\\nabla \\cdot(u\\nabla v)+\\mu u(1-u)-uz,\\\\ v_t=-(u+w)v,\\\\ w_t=\\Delta w-\\nabla \\cdot(w\\nabla v)-w+uz,\\\\ z_t=D_z\\Delta z-z-uz+\\beta w, \\end{array} \\right. \\end{equation*} in a smoothly bounded domain $\\Omega\\subset \\mathbb{R}^3$ with $\\beta>0$,~$\\mu>0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $\\beta \\max \\{1,\\|u_0\\|_{L^{\\infty}(\\Omega)}\\}<1+ (1+\\frac1{\\min_{x\\in \\Omega}u_0(x)})^{-1}$, this problem possesses a uniquely determined global classical solution $(u,v,w,z)$ for certain type of small data $(u_0,v_0,w_0,z_0)$. Moreover, $(u,v,w,z)$ is globally bounded and exponentially stabilizes towards its spatially homogeneous equilibrium %constant equilibrium $(1,0,0,0)$ as $t\\rightarrow \\infty$.","PeriodicalId":49860,"journal":{"name":"Mathematical Models & Methods in Applied Sciences","volume":" ","pages":""},"PeriodicalIF":3.6000,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy\",\"authors\":\"Yifu Wang, Chi Xu\",\"doi\":\"10.1142/s0218202523400043\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper deals with an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy \\\\begin{equation*} \\\\left\\\\{ \\\\begin{array}{lll} u_t=\\\\Delta u-\\\\nabla \\\\cdot(u\\\\nabla v)+\\\\mu u(1-u)-uz,\\\\\\\\ v_t=-(u+w)v,\\\\\\\\ w_t=\\\\Delta w-\\\\nabla \\\\cdot(w\\\\nabla v)-w+uz,\\\\\\\\ z_t=D_z\\\\Delta z-z-uz+\\\\beta w, \\\\end{array} \\\\right. \\\\end{equation*} in a smoothly bounded domain $\\\\Omega\\\\subset \\\\mathbb{R}^3$ with $\\\\beta>0$,~$\\\\mu>0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $\\\\beta \\\\max \\\\{1,\\\\|u_0\\\\|_{L^{\\\\infty}(\\\\Omega)}\\\\}<1+ (1+\\\\frac1{\\\\min_{x\\\\in \\\\Omega}u_0(x)})^{-1}$, this problem possesses a uniquely determined global classical solution $(u,v,w,z)$ for certain type of small data $(u_0,v_0,w_0,z_0)$. Moreover, $(u,v,w,z)$ is globally bounded and exponentially stabilizes towards its spatially homogeneous equilibrium %constant equilibrium $(1,0,0,0)$ as $t\\\\rightarrow \\\\infty$.\",\"PeriodicalId\":49860,\"journal\":{\"name\":\"Mathematical Models & Methods in Applied Sciences\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":3.6000,\"publicationDate\":\"2022-10-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Models & Methods in Applied Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0218202523400043\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Models & Methods in Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0218202523400043","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy
This paper deals with an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy \begin{equation*} \left\{ \begin{array}{lll} u_t=\Delta u-\nabla \cdot(u\nabla v)+\mu u(1-u)-uz,\\ v_t=-(u+w)v,\\ w_t=\Delta w-\nabla \cdot(w\nabla v)-w+uz,\\ z_t=D_z\Delta z-z-uz+\beta w, \end{array} \right. \end{equation*} in a smoothly bounded domain $\Omega\subset \mathbb{R}^3$ with $\beta>0$,~$\mu>0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $\beta \max \{1,\|u_0\|_{L^{\infty}(\Omega)}\}<1+ (1+\frac1{\min_{x\in \Omega}u_0(x)})^{-1}$, this problem possesses a uniquely determined global classical solution $(u,v,w,z)$ for certain type of small data $(u_0,v_0,w_0,z_0)$. Moreover, $(u,v,w,z)$ is globally bounded and exponentially stabilizes towards its spatially homogeneous equilibrium %constant equilibrium $(1,0,0,0)$ as $t\rightarrow \infty$.
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