  We consider parabolic partial differential equations of Lotka-Volterra type,
with a non-local nonlinear term. This models, at the population level, the
darwinian evolution of a population; the Laplace term represents mutations and
the nonlinear birth/death term represents competition leading to selection.
Once rescaled with a small diffusion, we prove that the solutions converge to a
moving Dirac mass. The velocity and weights cannot be obtained by a simple
expression, e.g., an ordinary differential equation. We show that they are
given by a constrained Hamilton-Jacobi equation. This extends several earlier
results to the parabolic case and to general nonlinearities. Technical new
ingredients are a $BV$ estimate in time on the non-local nonlinearity, a
characterization of the concentration point (in a monomorphic situation) and,
surprisingly, some counter-examples showing that jumps on the Dirac locations
are indeed possible.

Analysis of PDEs