  In this article, we perform an asymptotic analysis of a nonlocal
reaction-diffusion equation, with a fractional laplacian as the diffusion term
and with a nonlocal reaction term. Such equation models the evolutionary
dynamics of a phenotypically structured population. We perform a rescaling
considering large time and small effect of mutations, but still with algebraic
law. We prove that asymptotically the phenotypic distribution density
concentrates as a Dirac mass which evolves in time. This work extends an
approach based on Hamilton-Jacobi equations with constraint, that has been
developed to study models from evolutionary biology, to the case of fat-tailed
mutation kernels. However, unlike previous works within this approach, the WKB
transformation of the solution does not converge to a viscosity solution of a
Hamilton-Jacobi equation but to a viscosity supersolution of such equation
which is minimal in a certain class of supersolutions.
