Boundedness and exponential convergence in a chemotaxis model for tumor invasion

We revisit the following chemotaxis system modeling tumor invasion

in a smooth bounded domain $\Omega \subset {{\mathbb{R}}^{n}}(n\geqslant 1)$ with homogeneous Neumann boundary and initial conditions. This model was recently proposed by Fujie et al (2014 Adv. Math. Sci. Appl. 24 67–84) as a model for tumor invasion with the role of extracellular matrix incorporated, and was analyzed later by Fujie et al (2016 Discrete Contin. Dyn. Syst. 36 151–69), showing the uniform boundedness and convergence for $n\leqslant 3$. In this work, we first show that the ${{L}^{\infty}}$-boundedness of the system can be reduced to the boundedness of $\parallel u(\centerdot,t){{\parallel}_{{{L}^{\frac{n}{4}+\epsilon}}(\Omega )}}$ for some $\epsilon >0$ alone, and then, for $n\geqslant 4$, if the initial data $\parallel {{u}_{0}}{{\parallel}_{{{L}^{\frac{n}{4}}}}}$, $\parallel {{z}_{0}}{{\parallel}_{{{L}^{\frac{n}{2}}}}}$ and $\parallel \nabla {{v}_{0}}{{\parallel}_{{{L}^{n}}}}$ 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 Fujie et al (2016 Discrete Contin. Dyn. Syst. 36 151–69). More precisely, it is shown, if ${{u}_{0}}\geqslant,\not\equiv 0$, then any global and bounded solution (u, v, w, z) of the tumor invasion model satisfies the following exponential decay estimate: for all t  >  0 and for some constant C  >  0 independent of time t. Here, for a generic function f, $\bar{f}$ means the spatial average of f over $\Omega$ and ${{\lambda}_{1}}\left(>0\right)$ is the first nonzero eigenvalue of $- \Delta$ in $\Omega$ with homogeneous Neumann boundary condition.