| Abstract: |
| We consider a degenerate chemotaxis model for tumor invasion
in a bounded domain $\Omega \subset \mathbb{R}^N$ ($N\ge 2$),
which consists of four equations for $u, v, w, z$. The first equation
has the diffusivity $f$ and the sensitivity $g$ fulfilling
$f(u,w) \ge u^{m-1}$ ($m>1$), $0 \le g(u) \le u^\alpha$.
It is shown that if $\alpha+1 < m+\frac{4}{N}$ $(N \ge 2)$,
or if $\alpha+1=m+\frac{4}{N}$ $(N \ge 3)$ with small initial data,
then the system possesses a global bounded weak solution
which converges to the constant equilibrium in the weak$^*$ topology
in $L^\infty(\Omega)$ as $t\to\infty$. |
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