Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media
François Hamel, Grégoire Nadin · Proceedings of the American Mathematical Society · 2022
We consider in this article reaction-diffusion equations of the Fisher-KPP type with a nonlinearity depending on the space variable x x , oscillating slowly and non-periodically. We are interested in the width of the interface between the unstable steady state 0 0 and the stable steady state 1 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 → + ∞ t\rightarrow +\infty along some sequences of times, while it is sublinear along other sequences. As a corollary, under these conditions, generalized transition fronts do not exist for this equation.