Global solutions to a chemotaxis consumption model involving signal-dependent degenerate diffusion and logistic-type dampening
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 $Ω\subset \mathbb{R}^n,$ $n\geq1$, under no-flux boundary conditions, where the parameters $a,b>0$, $γ\geq2$, and the motility function $ϕ$ suitably generalizes the prototype given by $ϕ(s)=s^α$ for all $s\geq0$ with $α>0$. When $ϕ$ is appropriately smooth with $α\geq1$, it is shown that if one of the following cases holds: (i) $γ>2$; (ii) $γ=2$, either $n\leq2$ or $n\geq3$ and $b$ is sufficiently large, then for all suitably regular initial data global classical solutions can be constructed. Whereas when $ϕ$ is considered to be with rather mild regularity properties and $γ=2$, for arbitrary $b>0$, this system admits at least one global weak solution in case $α>0$. In addition, if $ϕ$ is suitably smooth with $α>1$, then the above weak solutions become eventually smooth.