Boundedness and exponential convergence of a chemotaxis model for tumor invasion

Abstract

We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=Δu-\nabla \cdot(u\nabla v),& x\inΩ, t>0,\\ v_t=Δv+wz,& x\inΩ, t>0,\\ w_t=-wz,& x\inΩ, t>0,\\ z_t=Δz-z+u, & x\inΩ, t>0,\\ \end{cases} \end{equation*} in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n(n\ge 1)$ 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 $L^{\infty }$-boundedness of the system can be reduced to the boundedness of $\Vert u(\cdot ,t){\Vert }_{L^{\frac{n}{4}+\epsilon }(\Omega )}$ for some $\epsilon >0$ alone, and then, for $n\ge 4$, if the initial data $\Vert u_0{\Vert }_{L^{\frac{n}{4}}}$, $\Vert z_0{\Vert }_{L^{\frac{n}{2}}}$ and $\Vert \nabla v_0{\Vert }_{L^n}$ 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}.

15pages, Submmitted

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