Boundedness and large-time behavior in a chemotaxis system with signal-dependent motility arising from tumor invasion

Abstract

In this paper, we study the following chemotaxis system with signal-dependent motility arising from tumor invasion

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta (\varphi (v)u)+\rho u -\mu u^l,&\qquad \quad x\in \Omega ,\,t>0,\\&v_t=\Delta v+ wz,&\qquad \quad x\in \Omega ,\,t>0,\\&w_t=-wz,&\qquad \quad x\in \Omega ,\,t>0,\\&z_t=\Delta z-z+u,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb {R}}^n(n\ge 1)$, where the parameters $\rho \ge 0,\mu \ge 0$ and $l>1$ are constants, the motility function $\varphi (v)$ satisfies $\varphi (v)\in C^3([0,+\infty )), \varphi _1\le \varphi (v)\le \varphi _2$ and $|\varphi '(v)|\le \varphi _3$ with $\varphi _1,\varphi _2,\varphi _3>0.$ The purpose of this paper is to prove that the existence of global bounded solution for $1\le n\le 3$. For $n\ge 4$, we prove that the nonnegative classical solution (u, v, w, z) is globally bounded if $l>\frac{n}{2}$. In addition, we also show that all the global bounded solution will converge to the non-trivial constant steady state exponentially.