natural question of characterizing which of the curves first touch, either $\S_D(c)$ and $\S_d(c)$ or $\tilde\S_D(c)$ and $\tilde\S_d(c)$. This analysis can be performed following the same ideas of the proof of Theorem \[thmain\](iii) (see Section \[sec8\]), i.e. by studying, for $c={c_{\operatorname{int}}}$, the second derivatives of the curves at $\b=0$, and adapting a fine result given in [@RTV Proposition 4.1], which is based on the comparison principles and characterizes the total number of possible intersections between the curves. Since such analysis is quite technical in general (indeed some of the arguments provided in Section \[sec8\] do not adapt to the general situation) and not strictly necessary for the results of this work, we send the reader to [@RTV] for the details.

Generalized subsolutions with compact support {#sec5}
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In this section we construct stationary compactly supported generalized (in the sense of Proposition \[prcp2\]) subsolutions in a framework moving in the $x-$direction at slightly smaller speeds than $c^*$, the one of Proposition \[pr41\]. Provided that Proposition \[prcp2\] can be applied, this will provide a lower bound for the asymptotic speed of propagation and will be the second and last ingredient for the proof of Theorem \[thmain\] (see Section \[sec6\]). The result is 