  In this paper we study the long time-long range behavior of reaction
diffusion equations with negative square root -type reaction terms. In
particular we investigate the exponential behavior of the solutions after a
standard hyperbolic scaling. This leads to a Hamilton-Jacobi variational
inequality with an obstacle that depends on the solution itself and defines the
open set where the limiting solution does not vanish. Counter-examples show a
nontrivial lack of uniqueness for the variational inequality depending on the
conditions imposed on the boundary of this open set. Both Dirichlet and state
constraints boundary conditions play a role. When the competition term does not
change sign, we can identify the limit, while, in general, we find lower and
upper bounds for the limit. Although models of this type are rather old and
extinction phenomena are as important as blow-up, our motivation comes from the
so-called "tail problem" in population biology. One way to avoid meaningless
exponential tails, is to impose extra-mortality below a given survival
threshold. Our study shows that the precise form of this extra-mortality term
is asymptotically irrelevant and that, in the survival zone, the population
profile is impacted by the survival threshold (except in the very particular
case when the competition term is non-positive).
