Minimal Wave Speed of Traveling Wavefronts in Delayed Belousov-Zhabotinskii Model

This paper is concerned with the traveling wavefronts of Belousov-Zhabotinskii model with time delay. By constructing upper and lower solutions and applying the theory of asymptotic spreading, the minimal wave speed is obtained under the weaker condition than that in the known results. Moreover, the strict monotonicity of any monotone traveling wavefronts is also established.

Recently, Lin and Li [5], Ma [6], Wu and Zou [11] further investigated the traveling wavefronts of the following delayed system where τ ≥ 0 is the time delay and other parameters are the same to those in (1.1 is not a monotone system in the sense of the standard partial ordering of the phase space of (1.3), we first make a change of variables such that we can study it by usual partial ordering when the theory of classical monotone dynamical systems is and (1.2) leads to Hereafter, a traveling wavefront of (1.4) is a special solution taking the following form In literature, the existence of (1.6)-(1.7) has been widely studied by many investigators. For example, when τ = 0, Troy [10] proved that for each b > 0, there exist r * > 0, c * ∈ (0, 2] such that (1.6)-(1.7) has a positive solution if we take c = c * , r = r * . However, the accurate presentation of c * , r * is not given and whether such a c * is the minimal wave speed is also open, here the minimal wave speed c * implies that (1.6)-(1.7) has no positive solution for c < c * while has a positive solution for c ≥ c * . Moreover, for τ = 0, Murray [8, pp.326] obtained the following bounds on minimal wave speed c in terms of r, b given by Furthermore, some results of existence of (1.6)-(1.7) with τ = 0 were obtained by constructing upper and lower solutions and it was proved that c 1 =: 2 √ 1 − r is the minimal wave speed if b ≤ 1 − r, we refer to Ye and Wang [12].
Recently, Ma [6], Wu and Zou [11] studied the existence of traveling wave solutions of (1.3) for c > c 1 and b ≤ 1 − r. Lin and Li [5] further proved the nonexistence of traveling wave solutions of (1.3) when the wave speed c < c 1 . At the same time, Lin and Li [5] also investigated the existence of (1.6) when c = c 1 holds, but the authors did not confirm (1.7). Therefore, even if b ≤ 1 − r, some problems of the existence/nonexistence of (1.6)-(1.7) remain open. Moreover, the role of time delay has not been reflected in these results, and we shall further consider the existence and nonexistence of (1.6)-(1.7).
In this paper, by constructing upper and lower solutions and utilizing the theory of asymptotic spreading, we prove that c 1 is the minimal wave speed of (1. which extends/completes some known results. In particular, it also confirms the conjecture of Lin and Li [5, Remark 2.4] by obtaining the asymptotic behavior of traveling wave solutions with c = c 1 . Moreover, we also consider the strict monotonicity of traveling wavefronts and prove the strict monotonicity of any monotone solutions of (1.6)-(1.7).

Preliminaries
In this section, we provide some known results such that we can follow our discussion in the subsequent section. Let X be defined by Using the notation, (1.6) equals to Let c > 0. Choose constants as follows Moreover, for (φ, ψ) ∈ X, define an operator which was earlier used by Wu and Zou [11]. Then, it is evident that a fixed point of F is a solution of (1.6) and it is sufficient to study the fixed point of F . In particular, we present a nice property of F as follows.
Lemma 2.1 is a special form of the comparison principle listed by [6,11] and we omit its proof here. To proceed our discussion, we need the following definition of upper and lower solutions (see [6,11]).
We now recall some conclusions of the so-called Fisher equation as follows in which d > 0 holds and w(x) is bounded and uniformly continuous.
Moreover, let w(x, t) = ρ(ξ), ξ = x + ct be a traveling wavefront of the following Fisher equation Then the following result is well known and can be found in many textbooks. We also state the following standard comparison principle.

Main Results
We first present our main result as follows.
We now prove Theorem 3.1 by five lemmas as follows.
(3.6) By (3.6) and Lemma 2.6, we see that Applying Lemma 2.4, we see that which implies a contradiction to (3.7) since By what we have done, we complete the proof. Summarizing these three lemmas, we obtain the minimal wave speed. Before continuing the paper, we also make a remark as follows.

Remark 3.5
Our results also answer the conjecture of Lin and Li [5,Remark 2.4] by confirming that α = 0.
The lemma is clear from the definition of F, and we omit it here.