Travelling waves for a non‐monotone bistable equation with delay: existence and oscillations. Issue 4 (24th November 2017) - Alfaro, Matthieu; Ducrot, Arnaud; Giletti, Thomas
Abstract: We consider a bistable ( 0 < θ < 1 being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in + ∞ . Combining refined a priori estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in − ∞ to 'something' which is strictly above the unstable equilibrium θ in + ∞ . Furthermore, we present situations (additional bound on the non‐linearity or small delay) where the wave converges to 1 in + ∞, whereas the wave is shown to oscillate around 1 in + ∞ when, typically, the delay is large.