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| We deal with the following chemotaxis system with non-diffusive chemical gradients arising from mathematical biology and medicine for $u:=u(x,t)$ and $v:=v(x,t)$
$\frac{\partial u}{\partial t}- \Delta u=-
\mbox{div}\left(\chi u \nabla v\right)+
\lambda u(1-u-v), (x,t) \in \Omega \times (0,T) $
$\frac{\partial v}{\partial t}+ h(v)=u,$
where $\Omega$ is a bounded domain in $R^n$ with the smooth boundary $\partial \Omega$, $\chi$ and
$\lambda $ are positive constants. We consider a intiial boundary value problem of the above system for the outer unit normal vector $n $
$\ \ \ \ \frac{\partial u }{\partial n}= \frac{\partial v }{\partial n}
=0 \qquad \mbox{in} \ \ \partial\Omega \times (0,T)$
$ \ \ \ \ u(x,0)=u_0(x), v(x,0)=v_0(x).$
We will discuss existence of global solutions in time and asymptotic behavior of solutions. |
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