Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

Abstract

We consider a bistable ($0\textless{}θ\textless{}1$ 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 $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $\theta$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large.

References in corpus (2)

• Travelling waves for a bistable reaction-diffusion equation with delay
• Existence and qualitative properties of travelling waves for an epidemiological model with mutations

Cited by in corpus (3)

• Sharp, Smooth, and Oscillatory Traveling Waves of Degenerate Diffusion Equation with Delay
• Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay
• On pushed wavefronts of monostable equation with unimodal delayed reaction