AI 中文总结
该研究针对具有退化扩散和紧支撑初始种群密度的趋化系统，证明了演化中出现的死核必源自初始零集，给出了死核形成的关键条件。
AI 中文摘要
我们考虑一类退化趋化系统，其形式为：\n\\[\n\left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\\\ &v_t=\Delta v+g(u,v),\end{array}\right.\n\\]\n该系统定义在有界区域\\(\Omega\subset\mathbb{R}^{N}\\)上，边界光滑，满足无通量和齐次Neumann边界条件。其中，扩散系数\\(D\in C^0([0,\infty))\cap C^1((0,\infty))\\)满足\\(D(0)=0\\)且在\\((0,\infty)\\)上\\(D'(s)\geq 0\\)；存在\\(s_0\in(0,1]\\)和\\(d>0\\)，使得在\\([0,s_0]\\)上\\(D(s)\geq ds^{m-1}\\)，且对\\(s\in[0,s_0]\\)有\\(s D'(s)\leq C_D D(s)\\)。第一个方程中的灵敏度函数\\(S\in C^2([0,\infty))\\)和源项\\(f\in C^{1}([0,\infty)\times[0,\infty))\\)均非负；第二个方程的源项\\(g\in C^{1}([0,\infty)\times[0,\infty))\\)可负，典型选择为\\(g(u,v)=-uv\\)或\\(g(u,v)=-v+u\\)。我们对\\(\Omega\times(0,T_0)\\)上的弱解施加适当假设后证明：若光滑有界区域\\(\omega\subset\mathbb{R}^N\\)和\\(T\in(0,T_0)\\)满足\\(\overline{\omega}\subseteq \Omega\\)、\\(u_0>0\\)在\\(\overline{\omega}\\)上、\\(u>0\\)在\\(\partial\omega\times(0,T)\\)上，则\\(u>0\\)在\\(\overline{\omega}\times[0,T)\\)上。特别地，演化过程中出现的任何死核都必然源自初始零集的已有区域。
英文摘要
We consider a degenerate chemotaxis system of the form \begin{align}\label{star}\tag{$\star$} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\ &v_t=Δv+g(u,v),\end{array}\right. \end{align} in a bounded domain $Ω\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^1((0,\infty))$ is assumed to satisfy $D(0)=0$ and $D'(s)\geq 0$ on $(0,\infty)$, and there are $s_0\in(0,1]$ and $d>0$ such that $D(s)\geq ds^{m-1}$ on $[0,s_0]$ and that \begin{align*} s D'(s)\leq C_D D(s)\quad\text{for }s\in[0,s_0]. \end{align*} The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ in the first equation are supposed to be nonnegative. The source term $g\in C^{1}([0,\infty)\times[0,\infty))$ of the second equation can in fact be negative. Prototypical choices for $g$ are $g(u,v)=-uv$ and $g(u,v)=-v+u$. We show under suitable assumptions on weak solutions to \eqref{star} on $Ω\times(0,T_0)$, that whenever the smoothly bounded domain $ω\subset\mathbb{R}^N$ and $T\in(0,T_0)$ are such that \begin{align*} \overlineω\subseteq Ω,\qquad u_0>0\ \text{ in }\ \overlineω,\qquad\text{ and }\qquad u>0\ \text{ on }\ \partialω\times(0,T), \end{align*} then \begin{align*} u>0\quad\text{in }\ \overlineω\times[0,T). \end{align*} In particular, any dead cores that appear during the evolution must have developed from regions that were already part of the initial zero set.
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