Boundedness in a three-dimensional chemotaxis–haptotaxis model

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• 作者：Xinru Cao
• 关键词：Chemotaxis ; Haptotaxis ; Logistic source ; Boundedness
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：67
• 期：1
• 全文大小：527 KB
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• 作者单位：Xinru Cao (1)
  
  1. Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

This paper studies the chemotaxis–haptotaxis system $$\left\{\begin{array}{lll} u_t = \Delta u - \chi\nabla \cdot (u\nabla v) - \xi\nabla \cdot (u\nabla w) + \mu u(1 - u - w), &\quad(x, t)\in \Omega \times (0, T),\\ v_t = \Delta v - v + u, &\quad(x, t) \in \Omega \times (0, T),\\ w_t= - vw, &\quad(x, t)\in \Omega \times (0,T)\end{array} \right.\quad\quad(\star)$$under Neumann boundary conditions. Here, \({\Omega \subset {\mathbb{R}}^3}\) is a bounded domain with smooth boundary and the parameters \({\xi,\chi,\mu > 0}\). We prove that for nonnegative and suitably smooth initial data \({(u_0, v_0, w_0)}\), if \({\chi/\mu}\) is sufficiently small, (\({\star}\)) possesses a global classical solution, which is bounded in \({\Omega \times (0, \infty)}\). We underline that the result fully parallels the corresponding parabolic–elliptic–ODE system. Keywords Chemotaxis Haptotaxis Logistic source Boundedness Mathematics Subject Classification 35B35 35B40 35K57 35Q92 92C17 Supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China.