  We derive a precise motion law for fronts of solutions to scalar
one-dimensional reaction-diffusion equations with equal depth multiple-wells,
in the case the second derivative of the potential vanishes at its minimizers.
We show that, renormalizing time in an algebraic way, the motion of fronts is
governed by a simple system of ordinary differential equations of nearest
neighbor interaction type. These interactions may be either attractive or
repulsive. Our results are not constrained by the possible occurrence of
collisions nor splittings. They present substantial differences with the
results obtained in the case the second derivative does not vanish at the
wells, a case which has been extensively studied in the literature, and where
fronts have been showed to move at exponentially small speed, with motion laws
which are not renormalizable.
