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 $\left(\begin{array}{cc}& u_t=\nabla \cdot (D\left(u\right)\nabla u-uS\left(u\right)\nabla v)+f(u,v),\\ & v_t=\Delta v-uv,\\ \end{array}\right)$in a bounded domain $\Omega \subset R^N$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0\left([0,\infty )\right)\cap C^2\left((0,\infty )\right)$ is assumed to satisfy $D\left(0\right)=0$, $D\left(s\right)>0$ on $(0,\infty )$, $D^{\prime }\left(s\right)\ge 0$ on $(0,\infty )$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $s{D}^{\prime }\left(s\right)\le C_{D}D\left(s\right)\text{and}{C}_D{s}^{p-1}\le D\left(s\right)\text{for}s\in [0,s_0].$The sensitivity function $S\in C^2\left([0,\infty )\right)$ 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 $v_0⁄\equiv 0$ there is a time-local classical solution and – despite the degeneracy at 0 – the solution satisfies an extensibility criterion of the form $\text{either}{T}_{max}=\infty ,\text{or}\underset{t\nearrow T_{max}}{lim\; sup}{\Vert u(\cdot ,t)\Vert }_{L^{\infty }\left(\Omega \right)}=\infty .$Moreover, as a by-product of our analysis, we prove that a classical solution on $\Omega \times (0,T)$ obeying ${\Vert u(\cdot ,t)\Vert }_{L^{\infty }\left(\Omega \right)}\le M_u$ 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 ${\delta }_u={\delta }_u(T,{\delta }_0,M_u,{\Vert v_0\Vert }_{W^{1,\infty }\left(\Omega \right)})>0$ 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.