Absence of dead-core formations in chemotaxis systems with degenerate diffusion

TL;DR

This paper proves that in a chemotaxis system with degenerate diffusion, dead-core formation cannot occur before potential blow-up, and it establishes conditions for the existence, positivity, and extensibility of solutions.

Contribution

It demonstrates the non-occurrence of dead-core formation in chemotaxis models with degenerate diffusion and provides criteria for solution existence and positivity.

Findings

Solutions exist globally or blow up in finite time.

Solutions remain strictly positive under certain conditions.

Dead-core formation is impossible before blow-up.

Abstract

In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form \begin{align*} \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- uv,\\ \end{array}\right. \end{align*} in a bounded domain $\Omega\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^2((0,\infty))$ is assumed to satisfy $D(0)=0$, $D(s)>0$ on $(0,\infty)$, $D'(s)\geq 0$ on $(0,\infty)$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $$s D'(s)\leq C_D D(s)\quad\text{and}\quad C_D s^{p-1}\leq D(s)\quad\text{for }s\in[0,s_0].$$ The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ are supposed to be nonnegative. We show that for all suitably regular initial data $(u_0,v_0)$ satisfying $u_0\geq \delta_0>0$ and $v_0\not\equiv 0$ there is a time-local classical solution and - despite the degeneracy at $0$ - the solution satisfies an extensibility criterion of the form $$\text{either}\quad T_{max}=\infty,\quad\text{or}\quad\limsup_{t\nearrow T_{max}}\|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty.$$ Moreover, as a by-product of our analysis, we prove that a classical solution on $\Omega\times(0,T)$ obeying $\|u(\cdot,t)\|_{L^\infty(\Omega)}\leq M_u$ for all $t\in(0,T)$ and emanating from initial data $(u_0,v_0)$ as specified above remains strictly positive throughout $\Omega\times(0,T)$, i.e. one can find $\delta_u=\delta_u(T,\delta_0, M_u,\|v_0\|_{W^{1,\infty}(\Omega)})>0$ such that $$u(x,t)\geq\delta_u\quad\text{for all }(x,t)\in\Omega\times(0,T).$$ Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time.