A simple approach to the wave uniqueness problem

Abstract

We propose a new approach for proving uniqueness of semi-wavefronts in generally nonmonotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

“…Again, we will distinguish between two following situations. In every case, (25) holds with ρ = min{ρ 1 , ρ 2 }.…”

“…On the other hand, φ(t) has no more than exponential rate of decay at −∞, cf. [25,Lemma 6]. Again, an application of [26,Lemma 22] shows that y(t) = v 0 (t)(1 + o(1)), t → +∞, where v 0 (t) is a non-zero eigensolution of the equation v (t) − cv (t) + v(t) = 0 corresponding to one of the positive eigenvalues z 1 , z 2 .…”

Nonstandard Quasi-monotonicity: An Application to the Wave Existence in a Neutral KPP–Fisher Equation

“…It is interesting to note that equation (3) with b = 0 coincides with the usual KPP-Fisher delayed equation, cf. [1,2,3,4,5,6,11,18,27]. There are at least four distinct demonstrations of the wavefront existence criterion for this equation (see [4,5,6,11]): our proof here differs from the previously known ones.…”

“…The proof of (B) is based on standard arguments from complex analysis (e.g. see [18,Appendix]) and it is omitted.…”

“…This approach, however, can be applied only to monotone waves and it also needs exact a priori estimates of wave asymptotics at +∞ (which seems to be more difficult to justify than similar estimates at −∞). Therefore, in this work, we are invoking a new idea recently proposed in [18] for functional diffusive equations with finite delays. The next theorem and its proof show that the mentioned idea in [18] is also significant for some equations with infinite delays.…”