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

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=Δv- 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 $(0,\infty )$, $D^{\prime }(s)\ge 0$ on $(0,\infty )$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $$s{D}^{\prime }(s)\le C_{D}D(s)\text{and}{C}_D{s}^{p-1}\le D(s)\text{for }s\in [0,s_0].$$ 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 $u_0\ge {\delta }_0>0$ 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 $$\text{either}{T}_{max}=\infty ,\text{or}\underset{t\nearrow T_{max}}{\mathrm{limsup}}\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}=\infty .$$ Moreover, as a by-product of our analysis, we prove that a classical solution on $\Omega \times (0,T)$ obeying $\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le M_u$ 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 ${\delta }_u={\delta }_u(T,{\delta }_0,M_u,\Vert v_0{\Vert }_{W^{1,\infty }(\Omega )})>0$ such that $$u(x,t)\ge {\delta }_u\text{for all }(x,t)\in \Omega \times (0,T).$$ 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.

7 pages

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