Traveling waves for a fisher-type reaction-diffusion equation with a flux in divergence form

Abstract

Abstract. Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction-di usion equations with ows in divergence form. We work with regular ows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic traveling waves move is an interval unbounded to the right. Contrary to classic examples, the in mum may not be reached. When the ow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic traveling wave speed threshold is complemented by another value by analyzing an extension of the rst order boundary value problem to which the classic case is reduced. This singular minimum speed can be justi ed as a viscous limit of classic minimal speeds in elliptic or over-elliptic ows. We construct a singular pro le for each speed between the minimum singular speed and the speeds at which classic traveling waves move. Under additional assumptions, the constructed pro le can be justi ed as that of a traveling wave of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine-Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.