Travelling waves for a non‐monotone bistable equation with delay: existence and oscillations. Issue 4 (24th November 2017)

• Record Type: Journal Article
• Authors: Alfaro, Matthieu; Ducrot, Arnaud; Giletti, Thomas
• Is Part Of: Proceedings of the London Mathematical Society. Volume 116:Issue 4(2018)
• Page Start: 729
• Page End: 759
• Publication Date: 2017-11-24
• Subjects: 35C07 -- 35K57 (primary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12092
• Languages: English
• ISSNs: 0024-6115

Abstract

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.