in a large class of reaction-diffusion equations, the solution starting from a compactly supported initial datum develops a transition between two rest states, that moves at an asymptotically linear rate in time, and whose thickness remains asymptotically bounded in time. the issue is its precise location in time, that is, up to terms that are o(1) as time goes to infinity. this question is well understood in one space dimension; i will discuss what happens in the less well settled multi-dimensional framework. joint works with l. rossi and v. roussier.