in $\ov{\O}$.

By Proposition \[prcp2\], the solution of with $\e(\un{u},\un{v})$ as initial datum, converges, increasingly, to a stationary solution $(U_2,V_2)$ of locally uniformly in $\overline{\O}$ and moreover $$U_2\leq\liminf_{t\to+\infty}u(x,t) \quad \text{ and } \quad V_2\leq\liminf_{t\to+\infty}v(x,y,t).$$ As before we have, for all $(x,y)\in\O$ $$U_2(x)>0, \quad V_{2,y}(x,0)>0, \quad V_2(x,y)>0 \quad \text{ and }  \quad V_2(x,L)>0$$ and, since $\e(\un{u},\un{v})$ is continuous and compactly supported and, thanks to the above-mentioned monotonicity, it does not touch $(U_2,V_2)$, we have that there exists $k>0$, $k\sim 0$, such that $\e(\un{u}(x-h),\un{v}(x-h,y))$, which still is a subsolution of lies below $(U_2(x),V_2(x,y))$ for all $h\in(-k,k)$. Anyway, by the uniqueness of the Cauchy problem associated to , the solution of with the translated subsolution as initial datum converges to the corresponding translation of $(U_2,V_2)$ and, by comparison, we have that $(U_2,V_2)$ is smaller than small translations in the $x$ direction of itself, which entails that the partial derivatives of $(U_2,V_2)$ with respect to $x$ are $0$.

We are now able to give the

*Proof of Theorem \[thltb\].* If holds, we obtain from Propositions \[pr31\] and \[pr32\], since Proposition \[prcp1\] guarantees that $(u,v)\geq(0,0)$. On the other hand, if and hold, follows from Propositions \[pr31\], \[pr32\] and \[pr34\].

Since we are interested in the 