  We consider in this article reaction-diffusion equations of the Fisher-KPP
type with a nonlinearity depending on the space variable x, oscillating slowly
and non-periodically. We are interested in the width of the interface between
the unstable steady state 0 and the stable steady state 1 of the solutions of
the Cauchy problem. We prove that, if the heterogeneity has large enough
oscillations, then the width of this interface, that is, the diameter of some
level sets, diverges linearly as t $\rightarrow$ +$\infty$ along some sequences
of times, while it is sublinear along other sequences. As a corollary, we show
that under these conditions generalized transition fronts do not exist for this
equation.
