Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay

Abstract

We study positive bounded wave solutions $u(t, x) = \phi(\nu \cdot x+ct)$, $\phi(-\infty)=0$, of equation $u_t(t, x)=\delta u(t,x)−u(t,x) +g(u(t−h,x))$, $x \in \mathbb{R}^m$(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: $u_1 \equiv 0$ and $u_2 \equiv \kappa>9$. The birth function $g \in C(\mathbb{R}_+, \mathbb{R}_+)$ satisfies a few mild conditions: it is unimodal and differentiable at $0,\kappa$. Some results also require the positive feedback of $g:[g(\text{max} g),\text{max} g] \to \mathbb{R}_+$ with respect to $\kappa$. If additionally $\phi(+\infty) =\kappa$, the above wave solution $u(t,x)$ is called a travelling front. We prove that every wave $\phi(\nu \cdot x+ct)$ is eventually monotone or slowly oscillating about $\kappa$. Furthermore, we indicate $c^∗ \in \mathbb{R}_+ \cup {+\infty}$ such that (∗) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity $c > c^∗$. Our results are based on a detailed geometric description of the wave profile $\phi$. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass $\mathcal{G}$ of ’asymmetric’ tent maps such that given $g \in \mathcal{G}$, there exists exactly one travelling front for each fixed admissible speed.