Boundedness in a quasilinear chemotaxis–haptotaxis model of parabolic–parabolic–ODE type

Abstract

Abstract This paper deals with the boundedness of solutions to the following quasilinear chemotaxis–haptotaxis model of parabolic–parabolic–ODE type: {ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−ur−1−w),x∈Ω,t>0,vt=Δv−v+uη,x∈Ω,t>0,wt=−vw,x∈Ω,t>0, $$ \textstyle\begin{cases} u_{t}=\nabla \cdot (D(u)\nabla u)-\chi \nabla \cdot (u\nabla v)- \xi \nabla \cdot (u\nabla w)+\mu u(1-u^{r-1}-w),& x\in \varOmega , t>0, \\ v_{t}=\Delta v-v+u^{\eta },& x\in \varOmega , t>0, \\ w_{t}=-vw, &x\in \varOmega , t>0, \end{cases} $$ under zero-flux boundary conditions in a smooth bounded domain Ω⊂Rn(n≥2) $\varOmega \subset \mathbb{R}^{n}(n\geq 2)$, with parameters r≥2 $r\geq 2$, η∈(0,1] $\eta \in (0,1]$ and the parameters χ>0 $\chi >0$, ξ>0 $\xi >0$, μ>0 $\mu >0$. The diffusivity D(u) $D(u)$ is assumed to satisfy D(u)≥δu−α $D(u)\geq \delta u^{-\alpha }$, D(0)>0 $D(0)>0$ for all u>0 $u>0$ with some α∈R $\alpha \in \mathbb{R}$ and δ>0 $\delta >0 $. It is proved that if α<n+2−2nη2+n $\alpha <\frac{n+2-2n\eta }{2+n}$, then, for sufficiently smooth initial data (u0,v0,w0) $(u_{0},v_{0},w_{0})$, the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded.

Keywords

• Chemotaxis
• Haptotaxis
• Nonlinear diffusion
• Boundedness
• Logistic source
• Nonlinear production