Global solutions to a chemotaxis consumption model involving signal-dependent degenerate diffusion and logistic-type dampening

Abstract

This work considers the Keller-Segel consumption system \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=Δ(uϕ(v))+au-bu^γ,\quad &x\in Ω,\quad t>0,\\ v_t=Δv-uv,\quad &x\inΩ,\quad t>0 \end{array} \right. \end{eqnarray*} in a smoothly bounded domain $\Omega \subset {\mathbb{R}}^n,$ $n\ge 1$, under no-flux boundary conditions, where the parameters $a,b>0$, $\gamma \ge 2$, and the motility function $\varphi$ suitably generalizes the prototype given by $\varphi (s)=s^{\alpha }$ for all $s\ge 0$ with $\alpha >0$. When $\varphi$ is appropriately smooth with $\alpha \ge 1$, it is shown that if one of the following cases holds: (i) $\gamma >2$; (ii) $\gamma =2$, either $n\le 2$ or $n\ge 3$ and $b$ is sufficiently large, then for all suitably regular initial data global classical solutions can be constructed. Whereas when $\varphi$ is considered to be with rather mild regularity properties and $\gamma =2$, for arbitrary $b>0$, this system admits at least one global weak solution in case $\alpha >0$. In addition, if $\varphi$ is suitably smooth with $\alpha >1$, then the above weak solutions become eventually smooth.

35 pages