Scholar - SciOpen

In this paper, we study a quasilinear chemotaxis model with a nonlinear indirect consumption mechanism

${\begin{cases} v_{1 t} = \nabla \cdot (ψ (v_{1}) \nabla v_{1} - χ ϕ (v_{1}) \nabla v_{2}) + λ_{1} v_{1} - λ_{2} v_{1}^{β}, & x \in Ω, t > 0, \\ v_{2 t} = Δ v_{2} - w^{θ} v_{2}, & x \in Ω, t > 0, \\ 0 = Δ w - w + v_{1}^{α}, & x \in Ω, t > 0, \end{cases}$

in a smooth and bounded domain $Ω \subset R^{n} (n \geq 1)$ with homogeneous Neumann boundary conditions, where $χ, λ_{1}, λ_{2}, θ > 0, 0 < α \leq \frac{1}{θ}, β \geq 2,$ $ψ$ , and $ϕ$ are nonlinear functions that satisfy $ψ (s) \geq a_{0} (s + 1)^{r_{1}}$ and $0 \leq ϕ (s) \leq b_{0} s (s + 1)^{r_{2}}$ for all $s \geq 0$ with $a_{0}, b_{0} > 0$ and $r_{1}, r_{2} \in R .$ It has been proven that if $r_{1} > 2 r_{2} + 1,$ then the problem admits a global and bounded classical solution for some appropriate nonnegative initial data.