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

In this paper, we study the following chemotaxis system with signal-dependent motility arising from tumor invasion \(\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta (\varphi (v)u)+\rho u -\mu u^l,&\qquad \quad x\in \Omega ,\,t>0,\\&v_t=\Delta v+ wz,&\qquad \quad x\in \Omega ,\,t>0,\\&w_t=-wz,&\qquad \quad x\in \Omega ,\,t>0,\\&z_t=\Delta z-z+u,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& u_t=\mathrm{\Delta }(\phi (v)u)+\rho u-\mu u^l,\hfill & \hfill x\in \mathrm{\Omega },t>0,\\ \hfill & v_t=\mathrm{\Delta }v+wz,\hfill & \hfill x\in \mathrm{\Omega },t>0,\\ \hfill & w_t=-wz,\hfill & \hfill x\in \mathrm{\Omega },t>0,\\ \hfill & z_t=\mathrm{\Delta }z-z+u,\hfill & \hfill x\in \mathrm{\Omega },t>0\end{array}\right)\end{array}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \( \Omega \subset {\mathbb {R}}^n(n\ge 1)\) $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ , where the parameters \( \rho \ge 0,\mu \ge 0\) $\rho \ge 0,\mu \ge 0$ and \( l>1\) $l>1$ are constants, the motility function \(\varphi (v)\) $\phi (v)$ satisfies \( \varphi (v)\in C^3([0,+\infty )), \varphi _1\le \varphi (v)\le \varphi _2\) $\phi (v)\in C^3([0,+\infty )),{\phi }_1\le \phi (v)\le {\phi }_2$ and \(|\varphi '(v)|\le \varphi _3\) $|{\phi }^{\prime }(v)|\le {\phi }_3$ with \(\varphi _1,\varphi _2,\varphi _3>0.\) ${\phi }_1,{\phi }_2,{\phi }_3>0.$ The purpose of this paper is to prove that the existence of global bounded solution for \(1\le n\le 3\) $1\le n\le 3$ . For \(n\ge 4\) $n\ge 4$ , we prove that the nonnegative classical solution (u, v, w, z) is globally bounded if \(l>\frac{n}{2}\) $l>\frac{n}{2}$ . In addition, we also show that all the global bounded solution will converge to the non-trivial constant steady state exponentially.