where $V$ satisfies and, thanks to the first equation of , $U=\frac{\nu}{\m} V(L)$. The first result regarding the long time behavior is the following

\[pr31\] Let $(u,v)$ be the solution of starting with a nonnegative, bounded initial datum $(u_0,v_0)$. Then, there exists a nonnegative, bounded solution $V_1$ of such that $$\limsup_{t\to+\infty}u(x,t)\leq U_1, \qquad \limsup_{t\to+\infty}v(x,y,t)\leq V_1(y)$$ locally uniformly in $\overline{\O}$, where $$U_1=\frac{\nu}{\m} V_1(L).$$

Observe preliminarily that, if we define, for $(x,y)\in\overline{\O}$, $$\overline{v}(x,y)=\max\left\{1,\left\|v_0\right\|_{\infty},\frac{\mu}{\nu}\left\|u_0\right\|_{\infty}\right\}, \qquad  \overline{u}(x)=\frac{\nu}{\mu}\overline{v},$$ then $(\overline{u},\overline{v})$ is a strict supersolution of which is larger than $(u_0,v_0)$. Therefore, by Proposition \[prcp1\], we have that the solution of with $(\overline{u},\overline{v})$ as initial datum is decreasing and, thanks to parabolic estimates, converges locally uniformly in $\overline{\O}$ to a nonnegative stationary solution $(U_1,V_1)$ of , i.e. a solution of . Proposition \[prcp1\] also gives $$\limsup_{t\to+\infty} u(x,t)\leq U_1(x), \qquad \limsup_{t\to+\infty} v(x,y,t)\leq V_1(x,y).$$ From the invariance of Problem in the $x$ direction and the uniqueness of the associated Cauchy problem, translations in $x$ of a solution of with a certain initial datum coincide with the solution of starting from the translated initial datum. Since $(\ov{u},\ov{v})$ is $x-$independent, the $x-$invariance of $(U_1,V_1)$ follows.

Obviously, admits the trivial solution $V=0$. In the following proposition we will show that is 