the following

\[pr51\] Let $c^*$ be the one constructed in Proposition \[pr41\]. Then, for every $c<c^*$, $c\sim c^*$, and $\d>0$, $\d\sim 0$, there exists $\e_0>0$ such that, for every $0<\e<\e_0$, $\e(\un{U}(x),\un{V}(x,y))$ is a compactly supported generalized (in the sense of Proposition \[prcp2\]) subsolution of $$\label{51}
  \left\{ \begin{array}{lll}
  u_t-Du_{xx}+c u_x=\nu v(x,L,t)-\m u & \text{for } x\in{{\mathbb{R}}}, & t>0 \\
  v_t-d\D v+c v_x=f(v) & \text{for } (x,y)\in\O, & t>0  \\ 
  dv_y(x,L,t)=\m u-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\
  v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0, \end{array}\right.$$ satisfying , where $(\un{U}(x),\un{V}(x,y))$ is a truncation of a solution of $$\label{52}
  \left\{ \begin{array}{ll}
  -D U_{xx}(x)+c U_x(x)=\nu V(x,L)-\m U(x) & \text{for } x\in{{\mathbb{R}}}, \\
  -d\D V(x,y)+c V_x(x,y)=(f'(0)-\d)V(x,y) & \text{for } (x,y)\in\O,   \\ 
  d V_y(x,L)=\m U(x)-\nu V(x,L) & \text{for } x\in{{\mathbb{R}}},  \\
  V(x,0)=0 & \text{for } x\in{{\mathbb{R}}}.  \end{array}\right.$$

Once the solution $(\un{U},\un{V})$ of will be constructed, the existence of $\e_0$ such that $\e(\un{U}(x),\un{V}(x,y))$ is a subsolution to for every $0<\e<\e_0$ follows immediately from . From the construction of $c^*$ carried out in the proof of Proposition \[pr41\], we know that, for $c<c^*$, no real solution of of type or exists. However we will show that exhibits, for $c<c^*$, $c\sim c^*$, complex solutions, which will be 