Stabilization in a chemotaxis model for tumor invasion

This paper deals with the chemotaxis system\[\begin{cases}u_t=\Delta u - \nabla \cdot (u\nabla v),\qquad x\in \Omega, \ t>0, \\v_t=\Delta v + wz,\qquad x\in \Omega, \ t>0, \\w_t=-wz,\qquad x\in \Omega, \ t>0, \\z_t=\Delta z - z + u, \qquad x\in \Omega, \ t>0,\end{cases}\]in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$,that has recently been proposed as a model for tumor invasionin which the role of an active extracellular matrix is accounted for. It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a correspondinginitial-boundary value problem of Neumann type possesses a global solution which is bounded.Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certainspatially homogeneous equilibrium in the sense that as $t\to\infty$, $u(x,t)\to \overline{u_0}$ ,  $v(x,t) \to \overline{v_0} + \overline{w_0}$,  $w(x,t) \to 0$   and   $z(x,t) \to \overline{u_0}$,  uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$,$\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$  and   $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.