  We consider a class of nonlocal reaction-diffusion problems, referred to as
replicator-mutator equations in evolutionary genetics. By using explicit
changes of unknown function, we show that they are equivalent to the heat
equation and, therefore, compute their solution explicitly. Based on this, we
then prove that, in the case of beneficial mutations in asexual populations,
solutions dramatically depend on the tails of the initial data: they can be
global, become extinct in finite time or, even, be defined for no positive
time. In the former case, we prove that solutions are accelerating, and in many
cases converge for large time to some universal Gaussian profile. This sheds
light on the biological relevance of such models.
