Avoidance of caldera-type dead cores in a chemotaxis system with degenerate diffusion and compactly supported initial population density
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.