  The notion of traveling wave, which typically refers to some particular
spatio-temporal con- nections between two stationary states (typically, entire
solutions keeping the same profile's shape through time), is essential in the
mathematical analysis of propagation phenomena. They provide insight on the
underlying dynamics, and an accurate description of large time behavior of
large classes of solutions, as we will see in this paper. For instance, in an
homogeneous framework, it is well-known that, given a fast decaying initial
datum (for instance, compactly supported), the solution of a KPP type
reaction-diffusion equation converges in both speed and shape to the traveling
wave with minimal speed. The issue at stake in this paper is the gener-
alization of this result to some one-dimensional heterogeneous environments,
namely spatially periodic or converging to a spatially periodic medium. This
result fairly improves our under- standing of the large-time behavior of
solutions, as well as of the role of heterogeneity, which has become a crucial
challenge in this field over the past few years.
