Combined effects of nonlinear diffusion and gradient-dependent flux limitation on a chemotaxis–haptotaxis model

Abstract

The flux-limited chemotaxis–haptotaxis system

$$\begin{aligned} \left\{ \begin{array}{l} \begin{aligned} &{}u_t = \nabla \cdot (D(u)\nabla u)-\chi \nabla \cdot (uf(|\nabla v|)\nabla v)-\xi \nabla \cdot (u\nabla w) +h(u,w), &{} x\in \Omega ,\ t>0,\\ &{}\tau v_t=\Delta v-v+u^{\eta }, &{} x\in \Omega ,\ t>0,\\ &{} w_t=-vw, &{}x\in \Omega ,\ t>0\\ \end{aligned} \end{array} \right. \end{aligned}$$

is considered in a smooth and bounded domain $\Omega \subset \mathbb R^n$ $(n\ge 2)$, where $\tau \in \{0,1\}$, $\chi ,\xi$ and $\eta$ are positive parameters, as well as the functions D, f and h satisfy $D(u)\ge d(u+1)^{m-1}$, $f(|\nabla v|)\le |\nabla v|^{p-2}$ and $h(u,w)=u(a-\mu u^{r-1}-\lambda w)$ with $a\in \mathbb R$, $d,\mu ,\lambda >0$, $r>1$, $r\ge 2\eta$, $m\ge 1$ and $1<p<\frac{n}{n-1}$. Firstly, for all reasonably regular initial data, we confirm the existence of a globally defined bounded classical solution if either $\tau =1$ and $n\in \{2,3\}$ or $\tau =0$ and $n\ge 2$. Moreover, when $\tau =0$ and alternatively $h(u,w)=a+au-\mu u^r-\lambda uw$ with $a>0$ and $r\ge 2$, it turns out that exponential decay of w is detected on large time scales, while both u and v persist in a certain manner for higher dimensions, provided that $\eta \ge 1$, $r\ge 2\eta$ and $\mu >\frac{\eta \chi ^2}{4r}$.