Stabilization in a chemotaxis model for tumor invasion
2015
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 invasion
in 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 corresponding
initial-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 certain
spatially 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   $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.
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