Boundedness and exponential convergence of a chemotaxis model for tumor invasion

We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=\Delta u-\nabla \cdot(u\nabla v),& x\in\Omega, t>0,\ v_t=\Delta v+wz,& x\in\Omega, t>0,\ w_t=-wz,& x\in\Omega, t>0,\ z_t=\Delta z-z+u, & x\in\Omega, t>0,\ \end{cases} \end{equation*} in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ with homogeneous Neumann boundary and initial conditions. This model was recently proposed by Fujie et al. \cite{FIY14} as a model for tumor invasion with the role of extracellular matrix incorporated, and was analyzed by Fujie et al. \cite{FIWY16}, showing the uniform boundedness and convergence for $n\le 3$. In this work, we first show that the $L^{\mathrm{\infty }}$-boundedness of the system can be reduced to the boundedness of $\mathrm{\parallel }u(\cdot ,t){\mathrm{\parallel }}_{L^{\frac{n}{4}+ϵ}(\mathrm{\Omega })}$ for some $ϵ>0$ alone, and then, for $n\ge 4$, if the initial data $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^{\frac{n}{4}}}$, $\mathrm{\parallel }{z}_0{\mathrm{\parallel }}_{L^{\frac{n}{2}}}$ and $|\nabla v_0 |_{L^n}$aresufficientlysmall,weareabletoestablishthe$L^\infty$-boundedness of the system. Furthermore, we show that boundedness implies exponential convergence with explicit convergence rate, which resolves the open problem left in \cite{FIWY16}.