  This paper is concerned with the large-time dynamics of bounded solutions of
reaction-diffusion equations with bounded or unbounded initial support in R N.
We start with a survey of some old and recent results on the spreading speeds
of the solutions and their asymptotic local one-dimensional symmetry. We then
derive some flattening properties of the level sets of the solutions if
initially supported on subgraphs. We also investigate the special case of
asymptotically conical-shaped initial conditions. Lastly, we reclaim some known
results about the logarithmic lag between the position of the solutions and
that of planar or spherical fronts expanding with minimal speed, for
almost-planar or compactly supported initial conditions. We then prove some new
logarithmic-in-time estimates of the lag of the position of the solutions with
respect to that of a planar front, for initial conditions which are supported
on subgraphs with logarithmic growth at infinity. These estimates entail in
particular that the same lag as for compactly supported initial data holds true
for a class of unbounded initial supports. The paper also contains some related
conjectures and open problems.
