Large time behavior of solution to a quasilinear chemotaxis model describing tumor angiogenesis with/without logistic source

Abstract

In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: $\left(\begin{array}{cc}{u}_t=\nabla \cdot (D\left(u\right)\nabla u)-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+\mu u-\mu u^2,& x\in \Omega ,t>0,\\ v_t=\Delta v+\nabla \cdot (v\nabla w)-v+u,& x\in \Omega ,t>0,\\ 0=\Delta w-w+u,& x\in \Omega ,t>0,\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0,& x\in \partial \Omega ,t>0,\\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),& x\in \Omega ,\end{array}\right)$in a bounded smooth domain $\Omega \subset R^n(n\le 3)$, where the parameter $\chi ,\xi >0,\mu \ge 0$, $D\left(u\right)$ is supposed to satisfy the behind property $D\left(u\right)\ge {(u+1)}^{\alpha }\text{with}\alpha >0.$Assume that either $\mu \ge 0,\alpha >1$ or $\mu =0,\xi \ge {\lambda }_1^{\ast }{\chi }^2$, where the parameter ${\lambda }_1^{\ast }={\lambda }_1^{\ast }(u_0,v_0,\Omega )>0$, then the system admits a global classical solution $(u,v,w)$ by subtle energy estimates. Moreover, when $\mu =0$, it is asserted that the corresponding solution exponentially converges to the constant stationary solution $(\overline{u_0},\overline{u_0},\overline{u_0})$ provided the initial data $u_0$ is sufficiently small, where $\overline{u_0}=\frac{{\int }_{\Omega }{u}_0}{\left|\Omega \right|}$. Finally, when $\mu >0$, it can be proved that the corresponding solution of the system decays to $(1,1,1)$ exponentially for suitable large $\mu$.