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 $\Omega \subset \mathbb{R}^n(n\geq 1)$ 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 $n\leq 3$. In this work, we first show that the $L^\infty$-boundedness of the system can be reduced to the boundedness of $\|u(\cdot,t)\|_{L^{\frac{n}{4}+\epsilon}(\Omega)}$ for some $\epsilon>0$ alone, and then, for $n\geq 4$, if the initial data $\|u_0\|_{L^{\frac{n}{4}}}$, $\|z_0\|_{L^\frac{n}{2}}$ and $\|\nabla v_0 \|_{L^n}$ are sufficiently small, we are able to establish the $L^\infty$-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}.