Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media

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