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We consider a class of nonlocal reaction-diffusion problems, referred to asreplicator-mutator equations in evolutionary genetics. By using explicitchanges of unknown function, we show that they are equivalent to the heatequation and, therefore, compute their solution explicitly. Based on this, wethen prove that, in the case of beneficial mutations in asexual populations,solutions dramatically depend on the tails of the initial data: they can beglobal, become extinct in finite time or, even, be defined for no positivetime. In the former case, we prove that solutions are accelerating, and in manycases converge for large time to some universal Gaussian profile. This shedslight on the biological relevance of such models.

Explicit solutions for replicator-mutator equations: extinction vs. acceleration