发表机构
Tokyo University of Science（东京科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究退化体积填充趋化系统带源项的初边值问题，在适当假设下证明全局弱解的存在性与唯一性。
AI 中文摘要
本文研究带有源项的退化体积填充趋化系统的无通量初边值问题：\\[ u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v) + f(x, u, v), \quad v_t = \Delta v + g(u,v), \quad x\in \Omega, \\ t>0 \\] 其中区域 $\Omega \subset \mathbb{R}^N$（$N \in \mathbb{N}$）光滑有界。文中证明：当 $D$、$h$、$f$ 和 $g$ 满足适当的假设，包括 $D(1,\cdot)=h(0,\cdot)=h(1,\cdot)=f(\cdot,0,\cdot) = f(\cdot,1,\cdot) = 0$ 时，对于满足 $u_0\le 1$ 的非负初值 $u_0$ 和 $v_0$，存在全局弱解 $(u, v)$ 且满足 $u\le 1$。此外，当 $D(r,s) = D(r)$ 对所有 $r\in[0,1]$ 和 $s\in[0,\infty)$ 成立，且 $D$、$h$、$f$、$g$ 和 $v_0$ 满足附加条件时，还建立了全局弱解的唯一性。
英文摘要
This paper is concerned with a no-flux initial-boundary value problem for the degenerate volume-filling chemotaxis system with source terms, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v) + f(x, u, v), \quad v_t = Δv + g(u,v), \quad x\in Ω, \ t>0 \end{align*} in a smoothly bounded domain $Ω\subset \mathbb{R}^N$ $(N \in \mathbb{N})$. It is shown that when $D$, $h$, $f$ and $g$ satisfy suitable assumptions involving $D(1,\cdot)=h(0,\cdot)=h(1,\cdot)=f(\cdot,0,\cdot) = f(\cdot,1,\cdot) = 0$, for nonnegative initial data $u_0$ and $v_0$ with $u_0\le 1$, there exists a global weak solution $(u, v)$ with $u\le 1$. In addition, uniqueness of global weak solutions is established when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$, and $D$, $h$, $f$, $g$ and $v_0$ are supposed to satisfy additional conditions.
CommentsChemotaxis, volume-filling, degenerate, existence, uniqueness
