Some counterexamples concerning a model of eradication of invasive species

In this note we study the regularity properties of the \emph{Effort Function $E(\beta )$}. This can be defined as the minimal $L^1$-norm of the control $\tilde{\alpha }(x)\ge 0$ so that the PDE $$U_t=\mathrm{\Delta }U+f(U)-\tilde{\alpha }(t,x)U,U\in [0,1],$$ admits a traveling profile $U(x-\beta t)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control $\tilde{\alpha }(t,x)$ describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of $f(U)$ and (consequently) on the effort function $E(\beta )$, the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that $E(\beta )$ is in general only $C^1$.