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

Abstract

We study positive bounded wave solutions u ( t , x ) = ϕ ( ν ⋅ x + c t ) , ϕ ( − ∞ ) = 0 , of equation u t ( t , x ) = Δ u ( t , x ) − u ( t , x ) + g ( u ( t − h , x ) ) , x ∈ R m ( ∗ ) . This equation is assumed to have two non-negative equilibria: u 1 ≡ 0 and u 2 ≡ κ > 0 . The birth function g ∈ C ( R + , R + ) is unimodal and differentiable at 0 and κ. Some results also require the feedback condition ( g ( s ) − κ ) ( s − κ ) < 0 , with s ∈ [ g ( max g ) , max g ] ∖ { κ } . If additionally ϕ ( + ∞ ) = κ , the above wave solution u ( t , x ) is called a travelling front. We prove that every wave ϕ ( ν ⋅ x + c t ) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c ∗ ∈ R + ∪ { + ∞ } such that Eq. ( ∗ ) 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 ϕ. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of ‘ asymmetric’ tent maps such that given g ∈ G , there exists exactly one positive travelling front for each fixed admissible speed.