Existence and Asymptotic Behavior of Traveling Wave Fronts for a Time-Delayed Degenerate Diffusion Equation

Abstract

This paper is concerned with traveling wave fronts for a degenerate diffusion equation with time delay. We first establish the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave fronts, respectively. Moreover, special attention is paid to the asymptotic behavior of traveling wave fronts connecting two uniform steady states. Some previous results are extended.

1. Introduction

In this paper, we consider the traveling wave fronts for the following reaction diffusion equation with Hodgkin-Huxley source: 𝜕𝑢=𝜕𝜕𝑡²𝑢^𝑚𝜕𝑥²+𝑓𝑢,𝑢_𝜏,𝑡≥0,𝑥∈ℝ, (1) where 𝑚>0, 𝑓(𝑢,𝑣)=𝑢^𝑝(1−𝑢)^𝑞(𝑣−𝑎), 𝑝>0, 𝑞>0, 𝑚+𝑝>1, 𝑎∈(0,1) is a constant, and 𝑢_𝜏(𝑥,𝑡)= 𝑢(𝑥,𝑡−𝜏) for 𝜏>0.

In 1952, Hodgkin and Huxley [1] proposed the Hodgkin-Huxley (H-H) equation 𝑑𝑢(𝑡(𝑡𝑢())(),𝑑𝑡=𝑢)(1−𝑢))(𝑡−𝜏−𝑎,𝑎∈0,1 (2) which describes the propagation of a voltage pulse through the nerve axon of a squid. Recently, more and more attention has been paid to the linear and semilinear parabolic equations with and without time delay; see, for example, [2–7]. A natural extension of the H-H model is the following linear diffusion equation: 𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑥²+𝑢(1−𝑢)𝑢_𝜏−𝑎. (3) For this equation, there have been many interesting results on the existence and stability of the traveling wave solutions, for instance, [8–10]. By a traveling wave solution, we mean a solution 𝑢(𝑥,𝑡) of (3) of the form 𝑢(𝑥,𝑡)=𝜑(𝑥+𝑐𝑡) with the wave speed 𝑐.

On the other hand, the classical research of traveling waves for the standard linear diffusion equations with various sources has been extended to some degenerate or singular diffusion equations. For example, Aronson [11] considered the following equation: 𝜕𝑢=𝜕𝜕𝑡²𝑢^𝑚𝜕𝑥²+𝑢(1−𝑢)(𝑢−𝑎),𝑎∈(0,1). (4) When 𝑚>1, the equation degenerates at 𝑢=0. Hence, it has a different feature from the case 𝑚=1; that is, if the initial distribution of 𝑢(𝑥,𝑡) has compact support, then 𝑢(𝑥,𝑡) also has compact support for each 𝑡>0. When 𝑚>1, 𝑎∈(0,1/2), Aronson [11] showed that (4) possesses a unique sharp traveling wave solution with positive wave speed. Hosono [12] solved the existence problem of traveling wave solutions for (4) especially with nonpositive wave speed and discussed the shape of the solutions. Sánchez-Garduño and Maini [13] considered the following degenerate diffusion equation: 𝜕𝑢=𝜕𝜕𝑡𝐷(𝑢)𝜕𝑥𝜕𝑢(),()(0)𝜕𝑥+𝑢1−𝑢)(𝑢−𝑎𝑎∈0,1,𝐷=0, (5) and obtained the existence of traveling wave solutions of smooth or sharp (oscillatory and monotone) type.

For other papers concerning the traveling wave solutions for degenerate diffusion equations without time delay, see [14–22]. From these results, we see that an obvious difference between the linear diffusion equations and the degenerate diffusion equations is that, in the degenerate diffusion case, there may exist traveling wave fronts of sharp type; that is, the support of the solution is bounded above or below, and at the boundary of the support, the derivative of the traveling wave solution is discontinuous. However, in the linear diffusion case, all traveling wave fronts are of smooth type; that is, the solutions are classical solutions, which approach the steady states at infinity.

As far as we know, there are only two articles dealing with the traveling wave solutions for degenerate diffusion equations with time delay. In [23, 24], Jin et al. considered the following time-delayed Newtonian filtration and non-Newtonian filtration equations: 𝜕𝑢=𝜕𝜕𝑡²𝑢^𝑚𝜕𝑥²+𝑢(1−𝑢)𝑢_𝜏−𝑎,𝜕𝑢=𝜕𝜕𝑡𝜕𝑥𝜕𝑢𝜕𝑥^𝑝−2𝜕𝑢𝜕𝑥+𝑢^𝑞(1−𝑢)𝑢_𝜏−𝑎. (6) By using the shooting method together with the comparison technique, they first obtained the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave solutions, respectively, and then gave an accurate estimation on the convergent rate for the semifinite or infinite traveling waves.

Motivated by [23, 24], in this paper, we discuss the existence and asymptotic behavior of traveling wave fronts for (1). Let 𝑢(𝑥,𝑡)=𝜑(𝜉) with 𝜉=𝑥+𝑐𝑡. Then, (1) is transformed into the following form: 𝑐𝜑^′(𝜉)=𝜑^𝑚′′(𝜉)+𝜑(𝜉)^𝑝1−𝜑(𝜉)^𝑞𝜑_𝑐𝜏(𝜉)−𝑎, (7) where 𝜑_𝑐𝜏(𝜉)=𝜑(𝜉−𝑐𝜏).

Before going further, we first give the definition of sharp- and smooth-type traveling wave fronts.

Definition 1. A function 𝜑(𝜉) is called a traveling wave front with wave speed 𝑐>0 if there exist 𝜉_𝑎,𝜉_𝑏 with −∞≤𝜉_𝑎<𝜉_𝑏≤+∞ such that 𝜑∈𝐶²(𝜉_𝑎,𝜉_𝑏) is monotonic increasing and 𝑐𝜑^′(𝜉)=𝜑^𝑚′′(𝜉)+𝜑(𝜉)^𝑝1−𝜑(𝜉)^𝑞𝜑_𝑐𝜏(𝜉),𝜉−𝑎𝜉∈_𝑎,𝜉_𝑏,𝜑(𝜉)=0for𝜉_𝑎−𝑐𝜏<𝜉≤𝜉_𝑎,𝜑𝜉_𝑏𝜑=1,_𝑚′𝜉_𝑎=𝜑^𝑚′𝜉_𝑏=0, (8)(9) or there exist 𝜉̃_𝑎, 𝜉̃_𝑏 with −∞≤𝜉̃_𝑎<𝜉̃_𝑏≤+∞ such that 𝜑∈𝐶²(𝜉̃_𝑎,𝜉̃_𝑏) is monotonic decreasing and 𝑐𝜑^′(𝜉)=𝜑^𝑚′′(𝜉)+𝜑(𝜉)^𝑝1−𝜑(𝜉)^𝑞𝜑_𝑐𝜏(𝜉),𝜉̃−𝑎𝜉∈_𝑎,𝜉̃_𝑏,𝜑(𝜉)=1for𝜉̃_𝑎−𝑐𝜏<𝜉≤𝜉̃_𝑎,𝜑𝜉̃_𝑏𝜑=0,_𝑚′𝜉̃_𝑎=𝜑^𝑚′𝜉̃_𝑏=0. (10)(11)(i)If 𝜉_𝑎>−∞ and 𝜑^′(𝜉⁺_𝑎)≠0, then 𝜑(𝜉) is called an increasing sharp-type traveling wave front (see Figure 1(a)). (ii)If 𝜉_𝑎=−∞ or 𝜑^′(𝜉⁺_𝑎)=0, then 𝜑(𝜉) is called an increasing smooth-type traveling wave front (see Figure 2(a)).  Similarly, we have the following.(iii)If 𝜉̃_𝑏<+∞ and 𝜑^′(𝜉̃⁻_𝑏)≠0, then 𝜑(𝜉) is called a decreasing sharp-type traveling wave front (see Figure 1(b)).(iv)If 𝜉̃_𝑏=+∞ or 𝜑^′(𝜉̃⁻_𝑏)=0, then 𝜑(𝜉) is called a decreasing smooth-type traveling wave front (see Figure 2(b)).

(a)

(b)

Figure 1 

Sharp-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing.

(a)

(b)

Figure 2 

Smooth-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing.

Let 𝜓(𝜉)=(𝜑^𝑚)^′(𝜉). Then, (8) or (10) is transformed into 𝜑^′(𝜉)=1𝑚𝜑^1−𝑚(𝜉)𝜓(𝜉),𝜓^′(𝜉)=𝑐𝑚𝜑^1−𝑚(𝜉)𝜓(𝜉)−𝜑(𝜉)^𝑝1−𝜑(𝜉)^𝑞𝜑_𝑐𝜏(𝜉)−𝑎. (12) Furthermore, by (9) or (11), we give the asymptotic boundary conditions for traveling wave fronts as follows: 𝜑(𝜉)=0for𝜉_𝑎−𝑐𝜏<𝜉≤𝜉_𝑎,𝜑𝜉_𝑏𝜓𝜉=1,_𝑎=𝜓𝜉_𝑏=0, (13) or 𝜑(𝜉)=1for𝜉̃_𝑎−𝑐𝜏<𝜉≤𝜉̃_𝑎,𝜑𝜉̃_𝑏𝜓𝜉̃=0,_𝑎=𝜓𝜉̃_𝑏=0. (14) If 𝜓(𝜉) is strictly positive or negative for 0<𝜑<1, then (12) is equivalent to 𝑑𝜓𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓. (15) Clearly, for any given 𝜑>0, if ^𝜑₀𝑚𝜎^𝑚−1𝜓(𝜎)𝑑𝜎=+∞, (16) then 𝜑_𝑐𝜏 can be defined by 𝑐𝜏=^𝜑_𝜑𝑐𝜏𝑚𝜎^𝑚−1𝜓(𝜎)𝑑𝜎. (17) However, if for some 𝜑>0, ^𝜑₀𝑚𝜎^𝑚−1𝜓(𝜎)𝑑𝜎<+∞, (18) then ∫^𝜑₀(𝑚𝜎^𝑚−1/𝜓(𝜎))𝑑𝜎 may be less than 𝑐𝜏 when 𝜑 is near 0. Therefore, the previous definition is not reasonable. In what follows, we give the definition of 𝜑_𝑐𝜏.

(i) If 𝜓 is positive, define 𝜑_𝑐𝜏 by 𝑐𝜏=^𝜑_𝜑𝑐𝜏𝑚𝜎^𝑚−1𝜓(𝜎)𝑑𝜎,if^𝜑₀𝑚𝜎^𝑚−1𝜓₊(𝜎)𝜑𝑑𝜎≥𝑐𝜏,_𝑐𝜏=0,if^𝜑₀𝑚𝜎^𝑚−1𝜓₊(𝜎)𝑑𝜎<𝑐𝜏, (19)

  where 𝜓₊ is a solution of the following problem: 𝑑𝜓𝑑𝜑=𝑐+𝑚𝑎𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜓(),𝜓0,for𝜑∈0,1₊𝜓𝜑().=0,>0,for𝜑∈0,1 (20) (ii) If 𝜓 is negative, define 𝜑_𝑐𝜏 by 𝑐𝜏=^𝜑𝑐𝜏_𝜑𝑚𝜎^𝑚−1−𝜓(𝜎)𝑑𝜎,if¹_𝜑𝑚𝜎^𝑚−1−𝜓₋(𝜎)𝜑𝑑𝜎≥𝑐𝜏,_𝑐𝜏=1,if¹_𝜑𝑚𝜎^𝑚−1−𝜓₋(𝜎)𝑑𝜎<𝑐𝜏, (21)

  where 𝜓₋ is a solution of the following problem: 𝑑𝜓𝑚()𝜑𝑑𝜑=𝑐−1−𝑎^{𝑚+𝑝−1}1−𝜑^𝑞𝜓(),𝜓(1,for𝜑∈0,1₋)𝜓𝜑().=0,<0,for𝜑∈0,1 (22) Consider the following problem: 𝑑𝜓𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓,𝜓0⁺=0,𝜓(1⁻)=0. (23) In Sections 2 and 3, we will verify the following two conclusions are equivalent, that is, (1) 𝜑 is a monotonic solution of the problem (8)-(9) (or (10)-(11)); (2) 𝜓(𝜑)>0 (or 𝜓(𝜑)<0) is a solution of the problem (23).

2. Existence of Increasing Traveling Waves

In this section, we aim to find a solution 𝜓(𝜑) of the problem (23) with 𝜓(𝜑)>0 for 𝜑∈(0,1).

Since 𝜓>0, we see that 𝜑(𝜉) is increasing in 𝜉, and so 𝜑_𝑐𝜏(𝜉)≤𝜑(𝜉). Thus, to investigate the behavior of the trajectories 𝜓_𝑐(𝜑) of (23), we have to study the trajectories starting from (0,0), since the property of 𝜓_𝑐 at 𝜑 depends on the behavior of 𝜓_𝑐 at 𝜑_𝑐𝜏 closely. Consider the following problem: 𝑑𝜓𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓,𝜓0for𝜑∈0,𝛽₊𝜓𝜑,=0,>0for𝜑∈0,𝛽 (24) where (0,𝛽) with 𝛽≤1 is the maximal existence interval of the solution 𝜓(𝜑)>0. By (24), 12𝜓²𝜑=^𝜑₀𝑐𝜓(𝑠)𝑑𝑠−𝑚^𝜑₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞𝑠_𝑐𝜏≥−𝑎𝑑𝑠_𝜑0𝑐𝜓(𝑠)𝑑𝑠−𝑚^𝜑₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠. (25) This excludes 𝜓(1⁻)=0 if ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≤0. Thus, we only need to find the increasing traveling wave fronts for the case ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠>0. We first prove the following two lemmas.

Lemma 2. Assume that 0<𝜑₀<1 and 𝜓₁(𝜑), 𝜓₂(𝜑) are solutions of (24) corresponding to different wave speeds 𝑐₁,𝑐₂, respectively, where  𝜓₁(𝜑₀)≥𝜓₂(𝜑₀)>0, 𝜓₁(𝜑)>𝜓₂(𝜑)>0 for 0<𝜑<𝜑₀, and 𝜑₁(𝜉₁)= 𝜑₂(𝜉₂)=𝜑₀. Then, 𝜑₁(𝜉₁−𝑠)<𝜑₂(𝜉₂−𝑠) if  0<𝑠<∫^𝜑0₀(𝑚𝜎^𝑚−1/𝜓₂(𝜎))𝑑𝜎 and 𝜑₁(𝜉₁−𝑠)= 𝜑₂(𝜉₂−𝑠)= 0 if 𝑠≥∫^𝜑0₀(𝑚𝜎^𝑚−1/𝜓₂(𝜎))𝑑𝜎.

Proof. Recalling (19), we see that 𝑠=^𝜑0_𝜑𝑖𝑠𝑚𝜎^𝑚−1𝜓_𝑖(𝜎)𝑑𝜎if^𝜑0₀𝑚𝜎^𝑚−1𝜓_𝑖(𝜎)𝜑𝑑𝜎>𝑠,_𝑖𝑠=0if^𝜑0₀𝑚𝜎^𝑚−1𝜓_𝑖(𝜎)𝑑𝜎≤𝑠, (26) where 𝜑_𝑖𝑠=𝜑_𝑖(𝜉_𝑖−𝑠),𝑖=1,2. The desired conclusion follows immediately.

Lemma 3. For any given 𝜏>0, and 𝑐₁>𝑐₂≥0, let 𝜓₁(𝜑), 𝜓₂(𝜑) be solutions of (24) corresponding to 𝑐₁, 𝑐₂, respectively. Then, 𝜓₁(𝜑)>𝜓₂(𝜑) for any 𝜑∈(0,𝛽₂), where (0,𝛽₂) is the maximal existence interval of the solution 𝜓₂(𝜑)>0. In addition, 𝜓₁(𝛽₂)>0. (See Figure 3(a).)

(a)

(b)

(c)

Figure 3 

The properties of the trajectory 𝜓𝑐(𝜑). (a) The monotonicity of 𝜓𝑐(𝜑) on 𝑐. (b) The trajectory 𝜓𝑐(𝜑) wanders through 𝐿𝑘. (c) The trajectory 𝜓𝑐(𝜑) intersects with 𝜓∗(𝜑) for large 𝑐.

Proof. We first show that 𝜓₁(𝜑)>𝜓₂(𝜑) for sufficiently small 𝜑>0. The argument consists of three cases, 𝑚+𝑝>2,𝑚+𝑝=2, and 1<𝑚+𝑝<2.
(i) Consider the case 𝑚+𝑝>2. According to (24), we have 𝑑𝜓𝑑𝜑=𝑐+𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝜓. (27) Noticing that 𝜑_𝑐𝜏≤𝜑, we have for 𝜑≤𝑎 that 𝑑𝜓𝑑𝜑≥𝑐. (28) Integrating from 0 to 𝜑 yields 𝜓𝜑≥𝑐𝜑. (29) We further have 𝑑𝜓𝑑𝜑=𝑐+𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝜓≤𝑐+𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝑐𝜑≤𝑐+𝑎𝑚𝑐𝜑^{𝑚+𝑝−2}. (30) Integrating from 0 to 𝜑 gives 𝜓𝜑≤𝑐𝜑+𝑎𝑚𝑐𝜑𝑚+𝑝−1^{𝑚+𝑝−1}. (31) That is, 𝜓𝜑∼𝑐𝜑as𝜑⟶0⁺. (32) Therefore, 𝜓₁(𝜑)>𝜓₂(𝜑) for sufficiently small 𝜑>0.
(ii) Consider the case 𝑚+𝑝=2. Similar to (i), we have 𝜓𝜑≥𝑐𝜑, (33) and, hence, 𝜓𝜑≤𝑐+𝑎𝑚𝑐𝜑. (34) Consider the sequences {𝐴_𝑛} and {𝐵_𝑛}, where 𝐴₁=𝑐,𝐵₁=𝑐+𝑎𝑚𝑐,𝐴_𝑛()=𝑐+1−𝜀^𝑞(𝑎−𝜀)𝑚𝐵_𝑛−1,𝐵_𝑛=𝑐+𝑎𝑚𝐴_𝑛−1,𝑛=2,3,…, (35) and 𝜀>0 is sufficiently small. Noticing that 𝐴₁<𝐵₁ and 𝐴₁<𝐴₂, by induction, we obtain 𝐴₁<𝐴₂=𝐴₃<𝐴₄=𝐴₅<⋯<𝐴_𝑛<⋯<𝐵_𝑛<⋯<𝐵₄=𝐵₃<𝐵₂=𝐵₁. (36) For 0<𝜑<𝜀, if 𝐴_𝑛−1𝜑≤𝜓𝜑≤𝐵_𝑛−1𝜑, (37) then 𝑑𝜓𝑑𝜑=𝑐+𝑚𝜑1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝜓≤𝑐+𝑚𝜑1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝐴_𝑛−1𝜑≤𝑐+𝑎𝑚𝐴_𝑛−1,𝑑𝜓𝑑𝜑=𝑐+𝑚𝜑1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝜓≥𝑐+𝑚𝜑1−𝜑^𝑞𝑎−𝜑_𝑐𝜏𝐵_𝑛−1𝜑()≥𝑐+1−𝜀^𝑞(𝑎−𝜀)𝑚𝐵_𝑛−1. (38) Integrating from 0 to 𝜑 yields 𝐴_𝑛𝜑≤𝜓𝜑≤𝐵_𝑛𝜑. (39) Thus, we have lim_𝑛→∞𝐴_𝑛=𝐴^∗(𝜀),lim_𝑛→∞𝐵_𝑛=𝐵^∗(𝜀),𝐴^∗(𝜀)𝜑≤𝜓𝜑≤𝐵^∗(𝜀)𝜑, (40) with 𝐴^∗(𝜀) and 𝐵^∗(𝜀) satisfying 𝐴^∗(𝜀)()=𝑐+1−𝜀^𝑞(𝑎−𝜀)𝑚𝐵^∗(𝜀),𝐵^∗(𝜀)=𝑐+𝑎𝑚𝐴^∗(𝜀). (41) Letting 𝜀→0, we obtain lim_𝜀→0𝐴^∗(𝜀)=lim_𝜀→0𝐵^∗(𝜀)=√𝑐²+4𝑚𝑎+𝑐2. (42) That is, 𝜓𝜑∼√𝑐²+4𝑚𝑎+𝑐2𝜑as𝜑⟶0⁺. (43) Therefore, 𝜓₁(𝜑)>𝜓₂(𝜑) for sufficiently small 𝜑>0.
(iii) Consider the case 1<𝑚+𝑝<2. Notice that 𝑑𝜓≥𝑑𝜑𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑𝜓, (44) which means that 𝜓²𝜑≥^𝜑₀2𝑚𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞()=𝑎−𝑠𝑑𝑠2𝑚𝑎𝜑𝑚+𝑝^𝑚+𝑝+𝑜𝜑^𝑚+𝑝; (45) that is, 𝜓𝜑≥2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+𝑜𝜑^{(𝑚+𝑝)/2}. (46) Consequently, 𝑑𝜓𝑑𝜑≤𝑐+𝑚𝑎𝜑^{𝑚+𝑝−1}𝜓≤𝑐+𝑚𝑎𝜑^{𝑚+𝑝−1}2𝑚𝑎/𝑚+𝑝𝜑^{(𝑚+𝑝)/2}+𝑜𝜑^{(𝑚+𝑝)/2}=𝑐+𝑚𝑎𝑚+𝑝2𝜑^{(𝑚+𝑝−2)/2}+𝑜𝜑^{(𝑚+𝑝−2)/2}. (47) That is, 𝜓𝜑≤𝑐𝜑+2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+𝑜𝜑^{(𝑚+𝑝)/2}. (48) Thus, we have 𝜓𝜑=2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+𝑜𝜑^{(𝑚+𝑝)/2}. (49) Recalling (46), we see that 𝑑𝜓²𝑑𝜑=2𝑐𝜓+2𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑_𝑐𝜏≥2𝑐2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+2𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑𝑎−𝜑+𝑜^{(𝑚+𝑝)/2}, (50) which implies that 𝜓²𝜑≥4𝑐𝑚+𝑝+22𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝+2)/2}+2𝑚𝑎𝜑𝑚+𝑝^𝑚+𝑝+𝑜𝜑^{(𝑚+𝑝+2)/2}. (51) On the other hand, by (49), we have 𝑑𝜓²𝑑𝜑≤2𝑐𝜓+2𝑚𝑎𝜑^{𝑚+𝑝−1}=2𝑐2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+2𝑚𝑎𝜑^{𝑚+𝑝−1}+𝑜𝜑^{(𝑚+𝑝)/2}, (52) and, hence, 𝜓²𝜑≤4𝑐𝑚+𝑝+22𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝+2)/2}+2𝑚𝑎𝜑𝑚+𝑝^𝑚+𝑝+𝑜𝜑^{(𝑚+𝑝+2)/2}. (53) Summing up, we arrive at 𝜓²𝜑=4𝑐𝑚+𝑝+22𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝+2)/2}+2𝑚𝑎𝜑𝑚+𝑝^𝑚+𝑝+𝑜𝜑^{(𝑚+𝑝+2)/2},as𝜑⟶0⁺, (54) which implies that 𝜓₁(𝜑)>𝜓₂(𝜑) for sufficiently small 𝜑>0.
We claim that 𝜓₁(𝜑)>𝜓₂(𝜑) for any 𝜑∈(0,𝛽₂). Suppose for contradiction that there exists 𝜑₀∈(0,𝛽₂) such that 𝜓₁(𝜑₀)=𝜓₂(𝜑₀)=𝜓₀ and 𝜓₁(𝜑)>𝜓₂(𝜑) for 𝜑∈(0,𝜑₀). Then, 𝜓^′₁𝜑₀≤𝜓^′₂𝜑₀, (55) which means that 𝑐₁+𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝑎−𝜑_0𝑐1𝜏𝜓₀≤𝑐₂+𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝑎−𝜑_0𝑐2𝜏𝜓₀; (56) that is, 0<𝑐₁−𝑐₂≤𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜓₀𝜑_0𝑐1𝜏−𝜑_0𝑐2𝜏. (57) Thus, 𝜑_0𝑐1𝜏>𝜑_0𝑐2𝜏. (58) Denote that 𝜑₁(𝜉₁)=𝜑₂(𝜉₂)=𝜑₀. By Lemma 2, we have 𝜑₁𝜉₁−𝑐₁𝜏≤𝜑₂𝜉₂−𝑐₁𝜏≤𝜑₂𝜉₂−𝑐₂𝜏, (59) which means that 𝜑_0𝑐1𝜏≤𝜑_0𝑐2𝜏, a contradiction.
In what follows, we will show that 𝜓₁(𝛽₂)>0. Recalling the first equation of (24), we infer that 12𝑑𝜓²_𝑖𝑑𝜑=𝑐_𝑖𝜓_𝑖−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝑖𝜏−𝑎,𝑖=1,2. (60) By Lemma 2, for any 𝜑₁(𝜉₁)=𝜑₂(𝜉₂)<𝛽₂, we have 𝜑₁𝜉₁−𝑐₁𝜏≤𝜑₂𝜉₂−𝑐₁𝜏≤𝜑₂𝜉₂−𝑐₂𝜏, (61) since 𝜓₁>𝜓₂. Thus, we obtain 𝑑𝜓²₁>𝑑𝜑𝑑𝜓²₂𝑑𝜑 (62) for any 𝜑∈(0,𝛽₂). Integrating the previous inequality from 𝜑₀ to 𝜑 for any 0<𝜑₀<𝜑<𝛽₂ yields 𝜓²₁𝜑>𝜓²₂𝜑+𝜓²₁𝜑₀−𝜓²₂𝜑₀. (63) Letting 𝜑→𝛽₂ gives 𝜓²₁𝛽₂≥𝜓²₁𝜑₀−𝜓²₂𝜑₀>0; (64) namely, 𝜓₁(𝛽₂)>0. The proof is completed.

To deal with the behavior of the trajectories 𝜓_𝑐(𝜑) of the problem (24), we introduce the level set 𝐿_𝑘=𝜑,𝜓∈ℝ²;12𝜓²+𝑚^𝜑₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠=𝑘 (65) for any 𝑘≥0. Clearly, for 𝑐=0, the level sets {𝐿_𝑘} are exactly the trajectories of solutions to (12) or (15). Now, we define 𝐿⁺_𝑘=𝐿_𝑘∩𝜓>0. (66) Notice that if (𝜑,𝜓)∈𝐿⁺_𝑘 is a solution of the system (12), we have 𝑑𝑘(𝜉)𝑑𝜉=𝜓𝜓^′+𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝜑^′=𝑐𝑚𝜑^1−𝑚𝜓²+𝜓𝜑^𝑝1−𝜑^𝑞𝜑−𝜑_𝑐𝜏≥𝑐𝑚𝜑^1−𝑚𝜓²>0, (67) since 𝜑(𝜉) is increasing in 𝜉. This implies that (𝜑,𝜓) wanders through increasing level sets with increasing 𝜉. See Figure 3(b). Let 𝑘₁=𝑚¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠; (68) we know that 𝐿_𝑘1 passes through the critical point (1,0). Denote that 𝜓^∗𝜑=2𝑚¹_𝜑𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠. (69) In what follows, we will see that 𝜓^∗(𝜑) plays a special role for the proof of the main result. We first need a lemma as follows.

Lemma 4. The trajectory 𝜓_𝑐(𝜑) of the problem (24) must intersect with  𝜓^∗(𝜑) for sufficiently large 𝑐>0. (See Figure 3(c).)

Proof. For any 𝜑∈(0,𝑎], we have 𝜑_𝑐𝜏∈(0,𝑎], and 𝜓^′_𝑐𝜑≥𝑐>0. (70) Let 𝜑₀∈(𝑎,1) be the first point such that 𝜓^′_𝑐(𝜑₀)=0. Then, we have 𝜓_𝑐𝜑₀=𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑_0𝑐𝜏−𝑎𝑐. (71) Since 𝜓_𝑐𝜑₀>𝜓_𝑐(𝑎)≥𝑎𝑐, (72) we have 𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑_0𝑐𝜏−𝑎>𝑎𝑐². (73) Thus, 𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑₀−𝑎>𝑎𝑐². (74) Denote that 𝑀=max_{𝜑∈(𝑎,1)}𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝑎. (75) Then, for any 𝑐≥√𝑀, (74) does not hold and 𝜓_𝑐(𝜑) is increasing on (𝑎,1). Therefore, 𝜓_𝑐(𝜑) must intersect with 𝜓^∗(𝜑) for any 𝑐≥√𝑀. The proof is completed.

Theorem 5. (i) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≤0, then there is no nontrivial nonnegative solution for problem (23).
(ii) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠>0, then there exists a unique 𝑐^∗₁>0, such that the problem (23) admits a nonnegative solution 𝜓(𝜑), and 𝜓(𝜑)>0 for any 𝜑∈(0,1).

Proof. (i) The case ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≤0 has been discussed.
(ii) We know that for any fixed 𝑐>0, (𝜑,𝜓_𝑐) wanders through increasing level sets {𝐿_𝑘} strictly. Thus, (𝜑,𝜓_𝑐) intersects with a level set 𝐿_𝑘 at most once. Let (𝜎_𝑐,𝜙_𝑐) be the intersection point of (𝜑,𝜓_𝑐) with 𝐿_𝑘1. Then, if 𝜙_𝑐>0, we have 𝑑𝜓^∗𝑑𝜑_{𝜑=𝜎𝑐}=−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝜓^∗_{𝜑=𝜎𝑐}≤−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜙_{𝑐𝜑=𝜎𝑐}=𝑑𝜓_𝑐𝑑𝜑_{𝜑=𝜎𝑐}−𝑐. (76) Define 𝑐^∗₁=inf𝑐>0;𝜓_𝑐intersects𝜓^∗at𝜎_𝑐,𝜙_𝑐with𝜎_𝑐∈(0,1]. (77) By Lemma 4, 𝑐^∗₁ is well defined. In what follows, we will show that 𝑐^∗₁ is just the desired wave speed.
We first have 𝑐^∗₁>0. Indeed, when 𝑐=0, the first equation of (24) becomes 𝑑𝜓₀𝑑𝜑=−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝜓₀. (78) Noticing that 𝜓₀(0⁺)=0, we have 𝜓²₀𝜑=−2𝑚^𝜑₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠. (79) Then, there exists 𝜑₀∈(𝑎,1) such that 𝜓₀𝜑₀=0, (80) since ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠>0. The continuous dependence of 𝜓_𝑐 on 𝑐 ensures that 𝜓_𝑐 goes to zero before reaching 𝜓^∗ for sufficiently small 𝑐>0. Thus, 𝑐^∗₁>0.
In addition, by Lemma 3, we know that 𝜎_𝑐 is decreasing on 𝑐. Assume that lim_{𝑐↘𝑐∗1}𝜎_𝑐=𝜎_𝑐^∗₁. (81) If 𝜎_𝑐^∗₁<1; namely, 𝜙_𝑐∗1>0, then by (76), we arrive at 𝜓^′_𝑐∗1𝜎_𝑐^∗₁≥𝜓^∗′𝜎_𝑐^∗₁+𝑐^∗₁>𝜓^∗′𝜎_𝑐^∗₁. (82) So, there exists 𝜎_𝑐^∗₁<𝜑₁<1, such that 𝜓_𝑐∗1(𝜑₁)>𝜓^∗(𝜑₁). By the continuous dependence of 𝜓_𝑐 on 𝑐, there is 𝑐>0 with 𝑐<𝑐^∗₁ and close to 𝑐^∗₁ sufficiently such that 𝜓_𝑐(𝜑₁)≥𝜓^∗(𝜑₁), which implies that 𝜓_𝑐 intersects with  𝜓^∗. This contradicts the definition of 𝑐^∗₁. Thus, 𝜎_𝑐^∗₁=1, which implies 𝜙_𝑐∗1=0.
Moreover, by Lemma 3, for 𝑐>𝑐^∗₁, we have 𝜓_𝑐(1)>0. However, if 0<𝑐<𝑐^∗₁, there exists 𝜑_𝑐 with 𝑎<𝜑_𝑐<1 such that 𝜓_𝑐(𝜑)→0 as 𝜑↗𝜑_𝑐. The proof is completed.

Proposition 6. 𝜑(𝜉) is a monotone increasing sharp- or smooth-type traveling wave front of the problem (8)-(9) for some fixed 𝑐>0, if and only if 𝜓(𝜑) with 𝜓(𝜑)>0 for any 𝜑∈(0,1) is a solution of the problem (23).

Proof. The necessity is clear. Consider the sufficient one. Let 𝜓(𝜑)>0 for any 𝜑∈(0,1) be a solution of the problem (23), and 𝜑(𝜉) solves 𝜑^′(𝜉)=1𝑚𝜑^1−𝑚(𝜉)𝜓𝜑(𝜉). (83) Without loss of generality, let 𝜑(0)=1/2, and let (𝛼,𝛽) be the maximal existence interval of 𝜑 such that 0<𝜑<1. Firstly, we have 𝜑^𝑚′′(𝜉)=𝜓^′𝜑𝜑^′(𝜉)=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓𝜑^′=𝑐𝜑^′−𝜑^𝑝1−𝜑^𝑞𝜑_𝑐𝜏−𝑎. (84) Moreover, 𝜑(𝜉)𝜑𝛽𝜑=0for𝛼−𝑐𝜏<𝜉≤𝛼,=1,^𝑚′(𝛼)=𝜑^𝑚′𝛽=0. (85) Therefore, if 𝛼>−∞ and 𝜑^′(𝛼⁺)≠0, 𝜑(𝜉) is a sharp-type traveling wave front; if 𝛼=−∞ or 𝜑^′(𝛼⁺)=0, 𝜑(𝜉) is a smooth-type traveling wave front.

Theorem 5 and Proposition 6 imply the following result.

Theorem 7. (i) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≤0, then there is no increasing traveling wave front for the problem (8)-(9).
(ii) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠>0, then there is a unique wave speed 𝑐^∗₁>0, such that the problem (8)-(9) admits an increasing traveling wave front.

Furthermore, we have the following results.

Theorem 8. Let 𝜑(𝜉) be the traveling wave front of the problem (8)-(9) corresponding to the wave speed 𝑐^∗₁.(i)If 𝑚<2, then 𝜑(𝜉) is a smooth-type traveling wave front. (ii)If 𝑚=2, then 𝜑^′(𝜉⁺_𝑎)=𝑐^∗₁/2 and 𝜑(𝜉) is a sharp-type traveling wave front. (iii)If 𝑚>2, then 𝜑^′(𝜉⁺_𝑎)=+∞ and 𝜑(𝜉) is a sharp-type traveling wave front.

Proof. (i) If 0<𝑚≤1, it is easy to see that 𝜑^′𝜉⁺_𝑎=1𝑚𝜑^1−𝑚𝜉⁺_𝑎𝜓𝜉⁺_𝑎=0. (86) If 1<𝑚<2, from the proof of Lemma 3, we see that when 𝜑>0 is sufficiently small, 𝜓𝜑=𝑐^∗₁𝜑𝜓𝜑=𝜑+𝑜,𝑚+𝑝>2;𝑐₁^∗2+4𝑚𝑎+𝑐^∗₁2𝜑𝜓𝜑=𝜑+𝑜,𝑚+𝑝=2;2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+𝑜𝜑^{(𝑚+𝑝)/2},1<𝑚+𝑝<2. (87) Since 𝜑^′(𝜉)=1𝑚𝜑^1−𝑚(𝜉)𝜓(𝜉), (88) we have 𝜑^′(𝜉)=𝑐^∗₁𝑚𝜑^2−𝑚(𝜉)+𝑜𝜑^2−𝑚(𝜉)𝜑,𝑚+𝑝>2;_′(𝜉)=𝑐₁^∗2+4𝑚𝑎+𝑐^∗₁𝜑2𝑚^2−𝑚(𝜉)+𝑜𝜑^2−𝑚(𝜉),𝜑𝑚+𝑝=2;_′(𝜉)=2𝑎𝑚𝜑𝑚+𝑝^{(2−𝑚+𝑝)/2}(𝜉)+𝑜𝜑^{(2−𝑚+𝑝)/2}(𝜉),1<𝑚+𝑝<2. (89) Thus, 𝜑^′𝜉⁺_𝑎=0. (90)
(ii) If 𝑚=2, we have 𝜑^′𝜉⁺_𝑎=𝑐^∗₁2. (91)
(iii) If 𝑚>2, we have 𝜑^′𝜉⁺_𝑎=+∞. (92) The proof is completed.

Theorem 9. 𝑐^∗₁(𝜏) is nonincreasing in delay 𝜏; namely, if 𝜏₁>𝜏₂, then 𝑐^∗₁(𝜏₁)≤𝑐^∗₁(𝜏₂).

Proof. Suppose for contradiction that there exist time delays 𝜏₁ and 𝜏₂, with 𝜏₁>𝜏₂ and 𝑐^∗₁(𝜏₁)>𝑐^∗₁(𝜏₂). Denote that 𝑐₁̂=𝑐^∗₁(𝜏₁) and  𝑐₂̂=𝑐^∗₁(𝜏₂) for simplicity. Take 𝑐 such that 𝑐₂̂<𝑐<𝑐₁̂. (93) In what follows, denote the solutions of (24) corresponding to wave speed and time delay (𝑐₁̂,𝜏₁), (𝑐,𝜏₁), (𝑐₂̂,𝜏₂) by 𝜓𝑐₁̂, 𝜓_𝑐, 𝜓𝑐₂̂, respectively.
Similar to the proof of Lemma 3, we know that when 𝜑>0 is sufficiently small, for 𝑚+𝑝>2, 𝜓_𝑐𝜑𝜑,𝜓𝑐̂=𝑐𝜑+𝑜₂𝜑=𝑐₂̂𝜑+𝑜𝜑; (94) for 𝑚+𝑝=2, 𝜓_𝑐𝜑=√𝑐²+4𝑚𝑎+𝑐2𝜑,𝜓𝑐̂𝜑+𝑜₂𝜑=𝑐²₂̂+4𝑚𝑎+𝑐₂̂2𝜑+𝑜𝜑; (95) for 1<𝑚+𝑝<2, 𝜓²_𝑐𝜑=4𝑐𝑚+𝑝+22𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝+2)/2}+2𝑚𝑎𝜑𝑚+𝑝^𝑚+𝑝+𝑜𝜑^{(𝑚+𝑝+2)/2},𝜓𝑐²₂̂𝜑=4𝑐₂̂𝑚+𝑝+22𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝+2)/2}+2𝑚𝑎𝜑𝑚+𝑝^𝑚+𝑝+𝑜𝜑^{(𝑚+𝑝+2)/2}. (96) Since 𝑐>𝑐₂̂, we have 𝜓_𝑐𝜑>𝜓𝑐₂̂𝜑 (97) for sufficiently small 𝜑>0. Furthermore, we claim that 𝜓_𝑐(𝜑)>𝜓𝑐₂̂(𝜑) for 𝜑∈(0,1). Otherwise, there exists 𝜑₀∈(0,1) such that 𝜓_𝑐(𝜑)>𝜓𝑐₂̂(𝜑) for 𝜑∈(0,𝜑₀), and 𝜓_𝑐(𝜑₀)=𝜓𝑐₂̂(𝜑₀)=𝜓₀. Then, we have 𝜓^′_𝑐𝜑₀≤𝜓𝑐^′₂̂𝜑₀; (98) that is, 0<𝑐−𝑐₂̂≤𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑_0𝑐𝜏1−𝜑₀𝑐̂_2𝜏2𝜓₀, (99) which implies 𝜑_0𝑐𝜏1>𝜑₀𝑐̂_2𝜏2. On the other hand, by Lemma 2, we have 𝜑_0𝑐𝜏1≤𝜑₀𝑐̂_2𝜏2, a contradiction. Similarly, we can get 𝜓𝑐₁̂(𝜑)>𝜓_𝑐(𝜑). From the uniqueness of wave speed on any fixed 𝜏>0, this contradicts the definition of 𝑐₁̂.

3. Existence of Decreasing Traveling Waves

In this section, we aim to find a solution 𝜓(𝜑) of the problem (23) with 𝜓(𝜑)<0 for 𝜑∈(0,1). We first introduce a comparison lemma.

Lemma 10. Let 𝑢,𝑣 be the solutions of the following problems, respectively: 𝑑𝑢𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝑢𝑓𝑜𝑟𝜑<1,𝑢(1⁻)=0,𝑑𝑣𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞(1−𝑎)𝑣𝑓𝑜𝑟𝜑<1,𝑣(1⁻)=0. (100) And let 𝜓 solve the problem (23). Then, 𝜓(𝜑)≤𝑢(𝜑)≤𝑣(𝜑) for 𝜑<1 if 𝜓, 𝑢, 𝑣 are positive, while 𝑣(𝜑)≤𝜓(𝜑)≤𝑢(𝜑) for 𝜑<1 if 𝜓, 𝑢, 𝑣 are negative.

Proof. Notice that 𝜑_𝑐𝜏≤𝜑 when 𝜓 is positive, and thus 12𝑑𝜓²𝑑𝜑−𝑐𝜓=−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎≥−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞=1𝜑−𝑎2𝑑𝑢²𝑑𝜑−𝑐𝑢; (101) that is, 𝑑𝜓²−𝑢²𝜓𝑑𝜑−2𝑐²−𝑢²𝑔𝜑≥0, (102) where 𝑔𝜑=1.𝜓+𝑢 (103) Consequently, 𝑑𝜓²−𝑢²𝑒^{2𝑐∫1𝜑𝑔(𝑠)𝑑𝑠}𝑑𝜑≥0. (104) Integrating from 𝜑 to 1 yields 𝜓²𝜑−𝑢²𝜑≤0, (105) and so 𝜓(𝜑)≤𝑢(𝜑) if 𝜓,𝑢>0. Similarly, 𝑢(𝜑)≤𝑣(𝜑) if 𝑢,𝑣>0.
The proof for 𝜓,𝑢,𝑣<0 is similar and omitted here.

Since 𝜓<0, we see that 𝜑(𝜉) is decreasing in 𝜉, and so 𝜑_𝑐𝜏(𝜉)≥𝜑(𝜉). To get the behavior of the trajectories 𝜓_𝑐(𝜑) of (23), we have to study the trajectories starting from (1,0), since the property of 𝜓_𝑐 at 𝜑 depends on the behavior of 𝜓_𝑐 at 𝜑_𝑐𝜏 closely. Consider the following problem: 𝑑𝜓𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓(),𝜓(1for𝜑∈𝛼,1₋)𝜓𝜑(),=0,<0for𝜑∈𝛼,1 (106) where (𝛼,1) with 𝛼≥0 is the maximal existence interval of the solution 𝜓(𝜑)<0. By (106), 12𝜓²𝜑=−¹_𝜑𝑐𝜓(𝑠)𝑑𝑠+𝑚¹_𝜑𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞𝑠_𝑐𝜏−𝑎𝑑𝑠≥−¹_𝜑𝑐𝜓(𝑠)𝑑𝑠+𝑚¹_𝜑𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠. (107) Hence, 𝜓(0⁺)=0 is impossible if ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≥0. So, we only need to find the decreasing traveling wave fronts for the case ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠<0. We first give the following three lemmas.

Lemma 11. Assume that 0<𝜑₀<1  and  𝜓₁(𝜑), 𝜓₂(𝜑) are solutions of (106) corresponding to different wave speeds 𝑐₁, 𝑐₂, respectively, where  𝜓₁(𝜑₀)≤𝜓₂(𝜑₀)<0, 𝜓₁(𝜑)<𝜓₂(𝜑)<0 for 𝜑₀<𝜑<1, and 𝜑₁(𝜉₁)=𝜑₂(𝜉₂)=𝜑₀. Then, 𝜑₁(𝜉₁−𝑠)>𝜑₂(𝜉₂−𝑠) if 0<𝑠<∫¹_𝜑0(𝑚𝜎^𝑚−1/−𝜓₂(𝜎))𝑑𝜎, and 𝜑₁(𝜉₁−𝑠)= 𝜑₂(𝜉₂−𝑠)=1 if 𝑠≥∫¹_𝜑0(𝑚𝜎^𝑚−1/−𝜓₂(𝜎))𝑑𝜎.

Proof. The proof is similar to that of Lemma 2.

Lemma 12. For any given 𝜏>0, and 𝑐₁>𝑐₂≥0, let 𝜓₁(𝜑), 𝜓₂(𝜑) be solutions of (106) corresponding to 𝑐₁, 𝑐₂, respectively. Then, 𝜓₁(𝜑)<𝜓₂(𝜑) for any 𝜑∈(𝛼₂,1), where (𝛼₂,1) is the maximal existence interval of the solution 𝜓₂(𝜑)<0. In addition, 𝜓₁(𝛼₂)<0. (See Figure 4(a).)

(a)

(b)

(c)

Figure 4 

The properties of the trajectory 𝜓𝑐(𝜑). (a) The monotonicity of 𝜓𝑐(𝜑) on 𝑐. (b) The trajectory 𝜓𝑐(𝜑) wanders through 𝑅𝑘. (c) The trajectory 𝜓𝑐(𝜑) intersects with 𝜓∗(𝜑) for large 𝑐.

Proof. We first show that 𝜓₁(𝜑)<𝜓₂(𝜑) if 𝜑 is in a sufficiently small left neighborhood of 1. The argument consists of three cases, 𝑞>1, 𝑞=1, and 𝑞<1.
(i) Consider the case 𝑞>1. According to (106), we have 𝑑𝜓𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓. (108) Noticing that 𝜑_𝑐𝜏≥𝜑, we have for 𝜑≥𝑎 that 𝑑𝜓𝑑𝜑≥𝑐. (109) Integrating from 𝜑 to 1 yields −𝜓𝜑≥𝑐1−𝜑. (110) We further have 𝑑𝜓𝑑𝜑=𝑐−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝜓≤𝑐+𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎𝑐𝑚()1−𝜑≤𝑐+1−𝑎𝑐1−𝜑^𝑞−1. (111) Integrating from 𝜑 to 1 gives 𝜑+𝑚()−𝜓≤𝑐1−𝜑1−𝑎𝑐𝑞1−𝜑^𝑞. (112) Thus, 𝜓𝜑∼𝑐𝜑−1,as𝜑⟶1⁻. (113) Therefore, 𝜓₁(𝜑)<𝜓₂(𝜑) in a left neighborhood of 𝜑=1.
(ii)   When 𝑞=1, consider the following two systems: 𝜑^′(𝜉)=1𝑚𝜑^1−𝑚(𝜉)𝑢₁(𝜉),𝑢^′₁(𝜉)=𝑐𝑚𝜑^1−𝑚(𝜉)𝑢₁(𝜉)−𝜑(𝜉)^𝑝(𝜉)𝜑(𝜉),𝜑1−𝜑−𝑎_′(𝜉)=1𝑚𝜑^1−𝑚(𝜉)𝑢₂(𝜉),𝑢^′₂(𝜉)=𝑐𝑚𝜑^1−𝑚(𝜉)𝑢₂(𝜉)−𝜑(𝜉)^𝑝1−𝜑(𝜉)(1−𝑎). (114) Notice that the right hand side of the previous two systems shares the same Jacobian matrix 01𝑃=𝑚𝑐1−𝑎𝑚 (115) at (1,0). By a simple calculation, we get the eigenvalues of the matrix 𝑃 as 𝜆₊=𝑐+𝑐²+4𝑚(1−𝑎)2𝑚,𝜆₋=𝑐−𝑐²+4𝑚(1−𝑎).2𝑚 (116) It is easy to see that (1,0) is a saddle point. And the eigenvector associated with the eigenvalue 𝜆₊ can be 1𝑄=𝑚𝜆₊. (117) We express the local explicit solutions of the problem (114) in the left neighborhood of (1,0) to reach 𝜑(𝜉)=1−𝑙𝑒^𝜆+𝜉+𝒪1−𝜑²,𝑢_𝑖(𝜉)=−𝑙𝑚𝜆₊𝑒^𝜆+𝜉+𝒪𝑢²_𝑖,𝑖=1,2. (118) Therefore, near (1,0), we have 𝑢_𝑖∼𝑚𝜆₊𝜑−1=𝑐+𝑐²+4𝑚(1−𝑎)2𝜑−1,𝑖=1,2. (119) By the comparison lemma, 𝑢₂≤𝜓≤𝑢₁, which implies 𝜓𝜑∼𝑐+𝑐²+4𝑚(1−𝑎)2𝜑−1,as𝜑⟶1⁻. (120) Thus, 𝜓₁(𝜑)<𝜓₂(𝜑) in a left neighborhood of 𝜑=1.
(iii) Consider the case 0<𝑞<1. Notice that 𝑑𝜓≥𝑑𝜑𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑𝜓, (121) which means that −𝜓²𝜑≤¹_𝜑2𝑚𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞()=()𝑎−𝑠𝑑𝑠2𝑚𝑎−1𝑞+11−𝜑^𝑞+1+𝑜1−𝜑^𝑞+1; (122) that is, −𝜓𝜑≥2𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+1)/2+𝑜1−𝜑^(𝑞+1)/2. (123) Consequently, 𝑑𝜓𝑚()𝑑𝜑≤𝑐−1−𝑎1−𝜑^𝑞𝜓𝑚()≤𝑐+1−𝑎1−𝜑^𝑞2𝑚(1−𝑎)/𝑞+11−𝜑^(𝑞+1)/2+𝑜1−𝜑^(𝑞+1)/2=𝑐+𝑚(1−𝑎)𝑞+121−𝜑^(𝑞−1)/2+𝑜1−𝜑^(𝑞−1)/2. (124) That is, −𝜓𝜑≤𝑐1−𝜑+2𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+1)/2+𝑜1−𝜑^(𝑞+1)/2. (125) Thus, we have −𝜓𝜑=2𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+1)/2+𝑜1−𝜑^(𝑞+1)/2. (126) Recalling (123), we see that −𝑑𝜓²𝑑𝜑=−2𝑐𝜓+2𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐𝜏−𝑎≥2𝑐2𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+1)/2+2𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎+𝑜1−𝜑^(𝑞+1)/2, (127) which implies that 𝜓²𝜑≥4𝑐𝑞+32𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+3)/2+2𝑚(1−𝑎)𝑞+11−𝜑^𝑞+1+𝑜1−𝜑^𝑞+1. (128) On the other hand, by (126), we have −𝑑𝜓²()𝑑𝜑≤−2𝑐𝜓+2𝑚1−𝑎1−𝜑^𝑞=2𝑐2𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+1)/2+2𝑚(1−𝑎)1−𝜑^𝑞+𝑜1−𝜑^(𝑞+1)/2, (129) and, hence, 𝜓²𝜑≤4𝑐𝑞+32𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+3)/2+2𝑚(1−𝑎)𝑞+11−𝜑^𝑞+1+𝑜1−𝜑^𝑞+1. (130) Summing up, we arrive at 𝜓²𝜑=4𝑐𝑞+32𝑚(1−𝑎)𝑞+11−𝜑^(𝑞+3)/2+2𝑚(1−𝑎)𝑞+11−𝜑^𝑞+1+𝑜1−𝜑^𝑞+1,as𝜑⟶1⁻, (131) which implies that 𝜓₁(𝜑)<𝜓₂(𝜑) in a left neighborhood of 𝜑=1.
The proof for the claims that 𝜓₁(𝜑)<𝜓₂(𝜑) for any 𝜑∈(𝛼₂,1) and 𝜓₁(𝛼₂)<0 is similar to the proof of Lemma 3 and omitted here.

Similar to Section 2, we denote level set 𝑅_𝑘 by 𝑅_𝑘=𝜑,𝜓∈ℝ²;12𝜓²−𝑚¹_𝜑𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠=𝑘 (132) for any 𝑘≥0, and correspondingly, define 𝑅⁻_𝑘=𝑅_𝑘∩𝜓<0. (133) Notice that if (𝜑,𝜓)∈𝑅⁻_𝑘 solves system (12), then 𝑑𝑘(𝜉)𝑑𝜉=𝜓𝜓^′+𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝜑^′=𝑐𝑚𝜑^1−𝑚𝜓²+𝜓𝜑^𝑝1−𝜑^𝑞𝜑−𝜑_𝑐𝜏≥𝑐𝑚𝜑^1−𝑚𝜓²>0, (134) since 𝜑(𝜉) is decreasing in 𝜉. This implies that the trajectory (𝜑,𝜓) of (106) wanders through increasing level sets with increasing 𝜉. See Figure 4(b). Letting 𝑘₂=−𝑚¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠, (135) we know that 𝑅_𝑘2 passes through the critical point (0,0). Denote that 𝜓^∗𝜑=−−2𝑚^𝜑₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠. (136) We introduce the following lemma.

Lemma 13. The trajectory 𝜓_𝑐(𝜑) of the problem (106) must intersect with  𝜓^∗(𝜑) for sufficiently large 𝑐>0. (See Figure 4(c).)

Proof. For any 𝜑∈[𝑎,1), we have 𝜑_𝑐𝜏∈[𝑎,1) and 𝜓^′_𝑐𝜑≥𝑐>0. (137) Let 𝜑₀∈(0,𝑎) be the first point such that 𝜓^′_𝑐(𝜑₀)=0. Then, we have 𝜓_𝑐𝜑₀=𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑_0𝑐𝜏−𝑎𝑐. (138) Since 𝜓_𝑐𝜑₀<𝜓_𝑐(𝑎)≤(𝑎−1)𝑐, (139) we have 𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑_0𝑐𝜏−𝑎<(𝑎−1)𝑐². (140) Thus, 𝑚𝜑₀^{𝑚+𝑝−1}1−𝜑₀^𝑞𝜑₀−𝑎<(𝑎−1)𝑐². (141) Denote that 𝑀=max_{𝜑∈(0,𝑎)}𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝑎−𝜑.1−𝑎 (142) Then, for any 𝑐≥𝑀, (141) does not hold, and 𝜓_𝑐(𝜑) is increasing on (0,𝑎). Therefore, 𝜓_𝑐(𝜑) must intersect with  𝜓^∗(𝜑) for any 𝑐≥𝑀. The proof is completed.

Theorem 14. (i) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≥0, then there is no nontrivial nonpositive solution for the problem (23).
(ii) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠<0, then there exists a unique 𝑐^∗₂>0, such that the problem (23) admits a nonpositive solution 𝜓(𝜑), and 𝜓(𝜑)<0 for any 𝜑∈(0,1).

Proof. The proof is similar to that of Theorem 5, and 𝑐^∗₂=inf𝑐>0;𝜓_𝑐intersects𝜓^∗at𝜂_𝑐,𝜙_𝑐with𝜂_𝑐∈[0,1). (143)

Proposition 15. 𝜑(𝜉) is a monotone decreasing sharp- or smooth-type traveling wave front of the problem (10)-(11) for some fixed 𝑐>0, if and only if 𝜓(𝜑) with 𝜓(𝜑)<0 for any 𝜑∈(0,1) is a solution of the problem (23).

Proof. The proof is similar to that of Proposition 6.

Theorem 14 and Proposition 15 imply the following result.

Theorem 16. (i) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠≥0, then there is no decreasing traveling wave front for the problem (10)-(11).
(ii) If ∫¹₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠<0, then there is a unique wave speed 𝑐^∗₂>0, such that the problem (10)-(11) admits a decreasing traveling wave front.

Furthermore, we have the following results.

Theorem 17. The traveling wave front 𝜑(𝜉) of the problem (10)-(11) corresponding to the wave speed 𝑐^∗₂ obtained earlier is of smooth type.

Proof. It is easy to see that 𝑑𝜓𝑑𝜑≤0,as𝜑⟶0⁺. (144) Thus, 𝜓≥𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐∗2𝜏−𝑎𝑐^∗₂≥−𝑚𝑎𝑐^∗₂𝜑^{𝑚+𝑝−1}. (145) Recalling that 𝜑^′(𝜉)=1𝑚𝜑^1−𝑚(𝜉)𝜓(𝜉), (146) we know that −𝑎𝑐^∗₂𝜑^𝑝(𝜉)≤𝜑^′(𝜉)≤0, (147) which means that 𝜑^′𝜉̃⁻_𝑏=0. (148) The proof is completed.

Theorem 18. 𝑐^∗₂(𝜏) is nonincreasing in delay 𝜏; namely, if 𝜏₁>𝜏₂, then 𝑐^∗₂(𝜏₁)≤𝑐^∗₂(𝜏₂).

Proof. The proof is similar to that of Theorem 9.

4. Asymptotic Behavior of the Traveling Wave Solutions

In this section, we first pay our attention to the finiteness of 𝜉_𝑎,𝜉_𝑏,𝜉̃_𝑎,𝜉̃_𝑏. On the basis of this, the convergent rates of 𝜑 going to the steady states at far-field are discussed.

Theorem 19. Let 𝜑(𝜉) be the increasing traveling wave solution of the problem (8)-(9) corresponding to the unique wave speed 𝑐^∗₁. (i)If0<𝑞<1, then 𝜉_𝑏<+∞. (ii)If 𝑞≥1, then 𝜉_𝑏=+∞. More precisely,(a)when 𝑞>1, one has (𝜉)∼()1−𝜑1−𝑎𝑞−1𝑐^∗₁𝜉^1/(1−𝑞),𝑎𝑠𝜉⟶+∞; (149)(b)when 𝑞=1, one has 1−𝜑(𝜉)∼𝑒^{−((𝑐1∗2+4𝑚(1−𝑎)−𝑐∗1)/2𝑚)𝜉},𝑎𝑠𝜉⟶+∞. (150)

Proof. (i) For (1+𝑎)/2≤𝜑_𝑐∗1𝜏<1, we have (1+𝑎)/2≤𝜑_𝑐∗1𝜏≤𝜑≤1, and 12𝑑𝜓²𝑑𝜑−𝑐^∗₁𝜓=−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐∗1𝜏𝑚()−𝑎≤−1−𝑎21+𝑎2^{𝑚+𝑝−1}1−𝜑^𝑞. (151) Consider the following inequality problem: 12𝑑𝑢²𝑑𝜑−𝑐^∗₁𝑚()𝑢≥−1−𝑎21+𝑎2^{𝑚+𝑝−1}1−𝜑^𝑞,𝑢(1for𝜑≤1,₋)𝑢𝜑=0,>0,for𝜑<1. (152) It is easy to prove that 𝜓(𝜑)≥𝑢(𝜑) when 𝜑_𝑐∗1𝜏≥(1+𝑎)/2. Now, we construct a function 𝑢 satisfying (152). Let 𝑢=𝐴1−𝜑^𝜃. (153) Then, 𝑢 satisfies (152) if and only if 𝐴²𝜃1−𝜑^{2𝜃−1−𝑞}+𝑐^∗₁𝐴1−𝜑^𝜃−𝑞≤𝑚(1−𝑎)21+𝑎2^{𝑚+𝑝−1}. (154) Take 𝜃=max1+𝑞2,𝑞. (155) Then, (154) holds when 𝐴 is appropriately small. Thus, 𝜓𝜑≥𝐴1−𝜑^𝜃, (156) with 𝜃=max{(1+𝑞)/2,𝑞} and 𝐴 appropriately small. Since 𝜑(𝜉) is the solution of the problem (8)-(9), there exists 𝜑₀<1 with 1−𝜑₀ small enough such that when 𝜑₀<𝜑<1, 𝜉𝜑−𝜉𝜑₀=^𝜑_𝜑0𝑚𝑠^𝑚−1𝜓(𝑠)≤𝑚𝑑𝑠𝐴^𝜑_𝜑0𝑠^𝑚−1(1−𝑠)^𝜃𝑑𝑠. (157) When 0<𝑞<1,𝜉(𝜑)<+∞ as 𝜑→1⁻. Thus, 𝜉_𝑏<+∞.
(ii) It is easy to see that 𝑑𝜓𝑑𝜑≤0,as𝜑⟶1⁻. (158) Thus, 𝜓≤𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐∗1𝜏−𝑎𝑐^∗₁≤𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑−𝑎𝑐^∗₁,𝜉𝜑−𝜉𝜑₀=^𝜑_𝜑0𝑚𝑠^𝑚−1𝜓(𝑠)≥𝑑𝑠_𝜑𝜑0𝑐^∗₁𝑠^𝑝(1−𝑠)^𝑞(𝑠−𝑎)𝑑𝑠. (159) Letting 𝜑→1, we obtain lim_𝜑→1−𝜉𝜑=+∞; (160) that is, 𝜉_𝑏=+∞. (161)
For any 𝜀>0, when 𝜑 approaches 1 enough, we have 𝜑_𝑐∗1𝜏>1−𝜀. Consider the following two problems: 12𝑑𝑢²𝑑𝜑−𝑐^∗₁𝑢≤−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞(),𝑢(11−𝑎for𝜑≤1,₋)𝑢𝜑1=0,>0for𝜑<1,2𝑑𝑣²𝑑𝜑−𝑐^∗₁𝑣≥−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞(),𝑣(11−𝜀−𝑎for𝜑≤1,₋)𝑣𝜑=0,>0for𝜑<1. (162)(163) It is easy to prove that 𝑣(𝜑)≤𝜓(𝜑)≤𝑢(𝜑) when 𝜑<1 with 𝜑_𝑐∗1𝜏>1−𝜀. In what follows, we construct two functions 𝑢 and 𝑣 satisfying (162) and (163), respectively. Let 𝑢=𝐴1−𝜑^𝜃. (164) Then, 𝑢 satisfies (162) if and only if 𝐴²𝜃1−𝜑^{2𝜃−1−𝑞}+𝑐^∗₁𝐴1−𝜑^𝜃−𝑞≥𝑚(1−𝑎)𝜑^{𝑚+𝑝−1}. (165) When 𝑞>1, take 𝑚()𝜃=𝑞,𝐴=1−𝑎𝑐^∗₁. (166) When 𝑞=1, take 𝜃=1,𝐴=𝑐₁^∗2+4𝑚(1−𝑎)−𝑐^∗₁2. (167) Then, 𝑢 is a solution of problem (162).
On the other hand, let 𝐴𝜃𝑣=1−𝜑. (168) Then, 𝑣 satisfies (163) if and only if 𝐴²𝜃1−𝜑²𝜃_−1−𝑞+𝑐^∗₁𝐴𝜃1−𝜑^−𝑞≤𝑚(1−𝜀−𝑎)𝜑^{𝑚+𝑝−1}. (169) When 𝑞>1, take 𝜃=𝑞; then, (169) is ensured by 𝐴²𝑞𝜀^𝑞−1+𝑐^∗₁𝐴≤𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}1−𝜀^(𝑞−1)/2+𝜀^(𝑞−1)/2, (170) which holds if 𝐴²𝑞𝜀^(𝑞−1)/2≤𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1},𝑐^∗₁𝐴≤𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}1−𝜀^(𝑞−1)/2. (171) Take 𝑚()𝐴=1−𝜀−𝑎)(1−𝜀^{𝑚+𝑝−1}1−𝜀^(𝑞−1)/2𝑐^∗₁. (172) Then, when 𝑐𝜀≤min1−𝑎,^{1∗4/(𝑞−1)}(𝑚(1−𝑎))^2/(𝑞−1)𝑞^2/(𝑞−1), (173) (169) holds. When 𝑞=1, take 𝜃=1, and 𝐴=𝑐₁^∗2+4𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}−𝑐^∗₁2. (174) Then, (169) is ensured.
Summing up, for any 𝜑<1 with 𝜑_𝑐∗1𝜏>1−𝜀, when 𝑞>1, 𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}1−𝜀^(𝑞−1)/2𝑐^∗₁1−𝜑^𝑞𝜑≤𝑚()≤𝜓1−𝑎𝑐^∗₁1−𝜑^𝑞. (175) Noticing that 𝜉𝜑=𝜉𝜑₀+^𝜑_𝜑0𝑚𝑠^𝑚−1𝜓(𝑠)𝑑𝑠, (176) we have 𝑐^∗₁(1−𝑎)𝑞−1≤lim_𝜑→1−𝜉𝜑1−𝜑^𝑞−1≤𝑐^∗₁(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}1−𝜀^(𝑞−1)/2;𝑞−1 (177) that is, (1−𝑎)𝑞−1𝑐^∗₁^1/(1−𝑞)≤lim_𝜉→+∞1−𝜑(𝜉)𝜉^1/(𝑞−1)≤(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}1−𝜀^(𝑞−1)/2𝑞−1𝑐^∗₁^1/(1−𝑞). (178) When 𝑞=1, 𝑐₁^∗2+4𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}−𝑐^∗₁2𝜑≤1−𝜑≤𝜓𝑐₁^∗2+4𝑚(1−𝑎)−𝑐^∗₁21−𝜑. (179) A direct calculation gives −2𝑚𝑐₁^∗2+4𝑚(1−𝑎)−𝑐^∗₁≤lim_𝜑→1−𝜉𝜑ln1−𝜑≤−2𝑚𝑐₁^∗2+4𝑚(1−𝜀−𝑎)(1−𝜀)^{𝑚+𝑝−1}−𝑐^∗₁; (180) that is, 𝑒^{−((𝑐1∗2+4𝑚(1−𝑎)−𝑐∗1)/2𝑚)𝜉}≤1−𝜑(𝜉)≤𝑒^{−((𝑐1∗2+4𝑚(1−𝜀−𝑎)(1−𝜀)𝑚+𝑝−1−𝑐∗1)/2𝑚)𝜉}. (181) By the arbitrariness of 𝜀>0, (149)-(150) hold.

Theorem 20. Let 𝜑(𝜉) be the increasing traveling wave solution of the problem (8)-(9) corresponding to the unique wave speed 𝑐^∗₁. (i)If 𝑚>min{𝑝,1}, then 𝜉_𝑎>−∞. (ii)If 𝑚≤min{𝑝,1}, then 𝜉_𝑎=−∞. More precisely, (a) when 𝑚+𝑝>2, if 𝑚=1, one has 𝜑(𝜉)∼𝑒^𝑐∗1𝜉,𝑎𝑠𝜉⟶−∞, (182) and if 𝑚<1, one has 𝜑(𝜉)∼(𝑚−1)𝑐^∗₁𝑚𝜉^1/(𝑚−1),𝑎𝑠𝜉⟶−∞; (183)(b)when 𝑚+𝑝=2, if 𝑚=1, one has 𝜑(𝜉)∼𝑒^{((𝑐1∗2+4𝑎+𝑐∗1)/2)𝜉},𝑎𝑠𝜉⟶−∞, (184) and if 𝑚<1, one has 𝜑(𝜉)∼(𝑚−1)𝑐₁^∗2+4𝑚𝑎+𝑐^∗₁𝜉2𝑚^1/(𝑚−1),𝑎𝑠𝜉⟶−∞; (185)(c)when 1<𝑚+𝑝<2, noticing that 𝑚≤min{𝑝,1}, then 𝑚≤𝑝; if 𝑚=𝑝, one has 𝜑(𝜉)∼𝑒^{(√𝑎/𝑚)𝜉},𝑎𝑠𝜉⟶−∞, (186) and if 𝑚<𝑝, one has 𝜑(𝜉)∼𝑚−𝑝2𝑚2𝑚𝑎𝜉𝑚+𝑝^{2/(𝑚−𝑝)},𝑎𝑠𝜉⟶−∞. (187)

Proof. For any 𝜑₀∈(0,𝑎), we see that 𝜉𝜑=𝜉𝜑₀−^𝜑0_𝜑𝑚𝑠^𝑚−1𝜓(𝑠)𝑑𝑠. (188) From the proof of Lemma 3, we see that when 𝜑>0 is sufficiently small, 𝜓𝜑=𝐴𝜑^𝛾+𝑜𝜑^𝛾, (189) where 𝛾=1 for 𝑚+𝑝≥2, and 𝛾=(𝑚+𝑝)/2 for 1<𝑚+𝑝<2. It is clear that 𝜉(𝜑)→−∞ as 𝜑→0⁺ if 𝛾≥𝑚, and 𝜉(𝜑) is finite as 𝜑→0⁺ if 𝛾<𝑚. That is, if 𝑚>min{𝑝,1}, then 𝜉_𝑎>−∞; however, if 𝑚≤min{𝑝,1}, then 𝜉_𝑎=−∞. Furthermore, notice that as 𝜑→0⁺, when 𝑚+𝑝>2, 𝜓𝜑=𝑐^∗₁𝜑+𝑜𝜑. (190) If 𝑚=1, we have lim_𝜑→0+𝜉𝜑=1ln𝜑𝑐^∗₁. (191) If 𝑚<1, lim_𝜑→0+𝜉𝜑𝜑^𝑚−1=𝑚(𝑚−1)𝑐^∗₁. (192) When 𝑚+𝑝=2, 𝜓𝜑=𝑐₁^∗2+4𝑚𝑎+𝑐2𝜑+𝑜𝜑. (193) If 𝑚=1, we have lim_𝜑→0+𝜉𝜑=ln𝜑𝑐₁^∗2+4𝑎−𝑐^∗₁.2𝑎 (194) If 𝑚<1, lim_𝜑→0+𝜉𝜑𝜑^𝑚−1=𝑐₁^∗2+4𝑚𝑎−𝑐^∗₁().2𝑎𝑚−1 (195) When 1<𝑚+𝑝<2, 𝜓𝜑=2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}+𝑜𝜑^{(𝑚+𝑝)/2}. (196) If 𝑚=𝑝, we have lim_𝜑→0+𝜉𝜑=𝑚ln𝜑√𝑎. (197) If 𝑚<𝑝, lim_𝜑→0+𝜉𝜑𝜑^{(𝑚−𝑝)/2}=2𝑚𝑚−𝑝𝑚+𝑝.2𝑚𝑎 (198) By a simple calculation, (182)–(187) hold.

Theorem 21. Let 𝜑(𝜉) be the decreasing traveling wave solution of the problem (10)-(11) corresponding to the unique wave speed 𝑐^∗₂. (i)If 0<𝑞<1, then 𝜉̃_𝑎>−∞. (ii)If 𝑞≥1, then 𝜉̃_𝑎=−∞. More precisely,(a)when 𝑞>1, one has 1−𝜑(𝜉)∼𝑒^{(𝑐∗2/𝑚)𝜉},𝑎𝑠𝜉⟶−∞; (199)(b)when 𝑞=1, one has 1−𝜑(𝜉)∼𝑒^{((𝑐2∗2+4𝑚(1−𝑎)+𝑐∗2)/2𝑚)𝜉},𝑎𝑠𝜉⟶−∞. (200)

Proof. Let 𝜑₀ approach 1 enough. Then, we have 𝜉𝜑₀−𝜉𝜑=^𝜑0_𝜑𝑚𝑠^𝑚−1𝜓(𝑠)𝑑𝑠. (201) From the proof of Lemma 12, we see that 𝜓𝜑∼𝐴1−𝜑^𝛾, (202) as 𝜑→1⁻, where 𝛾=1 for 𝑞≥1, and 𝛾=(𝑞+1)/2 for 0<𝑞<1. It is clear that 𝜉(𝜑)→−∞ as 𝜑→1⁻ if 𝛾≥1, and 𝜉(𝜑) is finite as 𝜑→1⁻ if 𝛾<1. That is, if 0<𝑞<1, then 𝜉̃_𝑎>−∞; however, if 𝑞≥1, then 𝜉̃_𝑎=−∞. Furthermore, notice that as 𝜑→1⁻, when 𝑞>1, 𝜓𝜑=𝑐^∗₂𝜑−1+𝑜𝜑−1,lim_𝜑→1−𝜉𝜑=𝑚ln1−𝜑𝑐^∗₂. (203) When 𝑞=1, 𝜓𝜑=𝑐₂^∗2+4𝑚(1−𝑎)+𝑐^∗₂2𝜑−1+𝑜𝜑−1,lim_𝜑→1−𝜉𝜑=ln1−𝜑2𝑚𝑐₂^∗2+4𝑚(1−𝑎)+𝑐^∗₂. (204) This yields (199)-(200).

Theorem 22. Let 𝜑(𝜉) be the decreasing traveling wave solution of the problem (10)-(11) corresponding to the unique wave speed 𝑐^∗₂. (i)If 𝑝<min{𝑚,1}, then 𝜉̃_𝑏<+∞. (ii)If 𝑝≥min{𝑚,1}, then 𝜉̃_𝑏=+∞. More precisely,(a)when 𝑚+𝑝>2, if 𝑝>1, one has 𝜑(𝜉)∼𝑎𝑝−1𝑐^∗₂𝜉^1/(1−𝑝),𝑎𝑠𝜉⟶+∞, (205) and if 𝑝=1, one has 𝜑(𝜉)∼𝑒^{−(𝑎/𝑐∗2)𝜉},𝑎𝑠𝜉⟶+∞; (206)(b)when 𝑚+𝑝=2, if 𝑝>1, one has 𝜑(𝜉)∼(1−𝑚)𝑐₂^∗2+4𝑚𝑎−𝑐^∗₂𝜉2𝑚^1/(𝑚−1),𝑎𝑠𝜉⟶+∞, (207) and if 𝑝=1, one has 𝜑(𝜉)∼𝑒^{−((𝑐2∗2+4𝑎−𝑐∗2)/2)𝜉},𝑎𝑠𝜉⟶+∞; (208)(c) when 1<𝑚+𝑝<2, noticing that 𝑝≥min{𝑚,1}, then 𝑝≥𝑚; if 𝑝>𝑚, one has 𝜑(𝜉)∼𝑝−𝑚2𝑚2𝑚𝑎𝜉𝑚+𝑝^{2/(𝑚−𝑝)},𝑎𝑠𝜉⟶+∞, (209) and if 𝑝=𝑚, one has 𝜑(𝜉)∼𝑒^{√−(𝑎/𝑚)𝜉},𝑎𝑠𝜉⟶+∞. (210)

Proof. (i) For 0<𝜑_𝑐∗2𝜏≤𝑎/2, we have 0≤𝜑≤𝜑_𝑐∗2𝜏≤𝑎/2, and 12𝑑𝜓²𝑑𝜑−𝑐^∗₂𝜓=−𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞𝜑_𝑐∗2𝜏≥−𝑎𝑚𝑎2𝑎1−2^𝑞𝜑^{𝑚+𝑝−1}. (211) Consider the following inequality problem: 12𝑑𝑢²𝑑𝜑−𝑐^∗₂𝑢≤𝑚𝑎2𝑎1−2^𝑞𝜑^{𝑚+𝑝−1}𝑢0,for𝜑≥0,₊𝑢𝜑=0,<0,for𝜑>0. (212) It is easy to prove that 𝜓(𝜑)≤𝑢(𝜑) when 𝜑_𝑐∗2𝜏≤𝑎/2. Now, we construct a function 𝑢 satisfying (212). Let 𝑢=−𝐵𝜑^𝑟. (213) Then, 𝑢 satisfies (212) if and only if 𝐵²𝑟𝜑^{2𝑟−𝑚−𝑝}+𝑐^∗₂𝐵𝜑^{𝑟−𝑚−𝑝+1}≤𝑚𝑎2𝑎1−2^𝑞. (214) Take 𝑟=max𝑚+𝑝2,𝑚+𝑝−1. (215) Then, (212) holds when 𝐵 is appropriately small. Thus, 𝜓𝜑≤−𝐵𝜑^𝑟, (216) with 𝑟=max{(𝑚+𝑝)/2,𝑚+𝑝−1} and 𝐵 appropriately small. Since 𝜑(𝜉) is the solution of the problem (10)-(11), there exists 𝜑₀>0 sufficiently small such that 𝜉𝜑−𝜉𝜑₀=−^𝜑0_𝜑𝑚𝑠^𝑚−1𝜓(𝑠)≤𝑚𝑑𝑠𝐵^𝜑0_𝜑1𝑠^{𝜃−𝑚+1}𝑑𝑠. (217) When 𝑝<min{𝑚,1}, 𝜉(𝜑)<+∞ as 𝜑→0⁺. Thus, 𝜉̃_𝑏<+∞.
(ii) From the proof of Theorem 14, we know that 𝜓𝜑≥𝜓^∗𝜑=−2𝑚^𝜑₀𝑠^{𝑚+𝑝−1}(1−𝑠)^𝑞(𝑎−𝑠)𝑑𝑠≥−2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}. (218) On the other hand, from the proof of Theorem 17, we also note that 𝜓𝜑≥−𝑚𝑎𝑐^∗₂𝜑^{𝑚+𝑝−1}. (219) By (218)-(219) and 𝜉𝜑−𝜉𝜑₀=^𝜑_𝜑0𝑚𝑠^𝑚−1𝜓(𝑠)𝑑𝑠, (220) it is easy to see that lim_𝜑→0+𝜉𝜑=+∞, (221) if 𝑝≥min{𝑚,1}. That is, 𝜉̃_𝑏=+∞. (222)
For any 0<𝜀<𝑎, when 𝜑 approaches 0 enough, we have 𝜑_𝑐∗2𝜏<𝜀. Consider the following problem: 12𝑑𝑢²𝑑𝜑−𝑐^∗₂𝑢≤𝑚𝜑^{𝑚+𝑝−1}1−𝜑^𝑞()𝑢0𝑎−𝜀,for𝜑≥0,₊𝑢𝜑=0,<0,for𝜑>0. (223) It is easy to prove that 𝜓(𝜑)≤𝑢(𝜑) when 𝜑>0 with 𝜑_𝑐∗2𝜏<𝜀. In what follows, we construct a function 𝑢 satisfying (223). Let 𝑢=−𝐵𝜑^𝑟. (224) Noticing that 0<𝜑≤𝜑_𝑐∗2𝜏<𝜀, then 𝑢 satisfies (223) if 𝐵²𝑟𝜑^{2𝑟−𝑚−𝑝}+𝑐^∗₂𝐵𝜑^{𝑟−𝑚−𝑝+1}≤𝑚(1−𝜀)^𝑞(𝑎−𝜀). (225) When 𝑚+𝑝>2, take 𝑟=𝑚+𝑝−1. Then, (225) is ensured by the following inequalities: 𝑐^∗₂𝐵≤𝑚(1−𝜀)^𝑞(𝑎−𝜀)1−𝜀^{(𝑚+𝑝−2)/2},𝑚+𝑝−1𝐵²𝑞𝜀^{(𝑚+𝑝−2)/2}≤𝑚(1−𝜀)^𝑞(𝑎−𝜀). (226) Take 𝑚()𝐵=1−𝜀^𝑞(𝑎−𝜀)1−𝜀^{(𝑚+𝑝−2)/2}𝑐^∗₂. (227) Then, when 𝜀<(𝑐₂^∗2/𝑚(𝑚+𝑝−1))^{2/(𝑚+𝑝−2)}, (225) holds. Recalling (219), we obtain −𝑚𝑎𝑐^∗₂𝜑^{𝑚+𝑝−1}𝜑𝑚()≤𝜓≤−1−𝜀^𝑞(𝑎−𝜀)1−𝜀^{(𝑚+𝑝−2)/2}𝑐^∗₂𝜑^{𝑚+𝑝−1} (228) when 𝜑_𝑐∗2𝜏<𝜀. When 𝑝>1, we have 𝑐^∗₂(1−𝜀)^𝑞(𝑎−𝜀)1−𝜀^{(𝑚+𝑝−2)/2}𝑝−1≤lim_𝜑→0+𝜉𝜑𝜑^𝑝−1≤𝑐^∗₂𝑎;𝑝−1 (229) that is, (1−𝜀)^𝑞(𝑎−𝜀)1−𝜀^{(𝑚+𝑝−2)/2}𝑝−1𝑐^∗₂^1/(1−𝑝)≤lim_𝜉→+∞𝜑(𝜉)𝜉^1/(𝑝−1)≤𝑎𝑝−1𝑐^∗₂^1/(1−𝑝). (230) When 𝑝=1, we have −𝑐^∗₂(1−𝜀)^𝑞(𝑎−𝜀)1−𝜀^{(𝑚+𝑝−2)/2}≤lim_𝜑→0+𝜉𝜑𝑐ln𝜑≤−^∗2𝑎; (231) that is, 𝑒^{−(𝑎/𝑐∗2)𝜉}≤𝜑(𝜉)≤𝑒^{−((1−𝜀)𝑞(𝑎−𝜀)(1−𝜀(𝑚+𝑝−2)/2)/𝑐∗2)𝜉}. (232) By the arbitrariness of 𝜀>0, (205)-(206) hold.
When 1<𝑚+𝑝<2, take 𝑟=(𝑚+𝑝)/2. Then, (225) holds if 𝑚+𝑝2𝐵²≤𝑚(1−𝜀)^𝑞(𝑎−𝜀)1−𝜀^{(2−𝑚−𝑝)/4},𝑐^∗₂𝐵𝜀^{(2−𝑚−𝑝)/4}≤𝑚(1−𝜀)^𝑞(𝑎−𝜀). (233) Take 𝐵=2𝑚()𝑚+𝑝1−𝜀^𝑞(𝑎−𝜀)1−𝜀^{(2−𝑚−𝑝)/4}. (234) Then, when 𝑎𝜀≤min2,𝑎2𝑎1−2^{𝑞2/(2−𝑚−𝑝)}×𝑚𝑚+𝑝2𝑐₂^{∗−22/(2−𝑚−𝑝)}, (235) (225) holds. Combining with (218), we obtain −2𝑚𝑎𝜑𝑚+𝑝^{(𝑚+𝑝)/2}≤𝜓𝜑≤−2𝑚()𝑚+𝑝1−𝜀^𝑞(𝑎−𝜀)1−𝜀^{(2−𝑚−𝑝)/4}𝜑^{(𝑚+𝑝)/2} (236) when 𝜑_𝑐∗2𝜏<𝜀. When 𝑝>𝑚, a simple calculation yields 𝑝−𝑚2𝑚2𝑚(1−𝜀)^𝑞(𝑎−𝜀)1−𝜀^{(2−𝑚−𝑝)/4}𝑚+𝑝^{2/(𝑚−𝑝)}≤lim_𝜉→+∞𝜑(𝜉)𝜉^{2/(𝑝−𝑚)}≤𝑝−𝑚2𝑚2𝑚𝑎𝑚+𝑝^{2/(𝑚−𝑝)}. (237) When 𝑝=𝑚, 𝑒^{√−(𝑎/𝑚)𝜉}≤𝜑(𝜉)≤𝑒^{−((1−𝜀)𝑞(𝑎−𝜀)(1−𝜀(1−𝑚)/2)/𝑚)𝜉}. (238) By the arbitrariness of 𝜀>0, (209)-(210) hold.
When 𝑚+𝑝=2, take 𝑟=1,𝐵=𝑐₂^∗2+4𝑚(1−𝜀)^𝑞(𝑎−𝜀)−𝑐^∗₂2. (239) Then, (225) holds. On the other hand, consider the following problem: 12𝑑𝑣²𝑑𝜑−𝑐^∗₂𝑣≥𝑚𝑎𝜑1−𝜑^𝑞𝑣0,for𝜑≥0,₊𝑣𝜑=0,<0,for𝜑>0. (240) We can see that 𝜓(𝜑)≥𝑣(𝜑). Let 𝑣=−𝑐₂^∗2+4𝑚𝑎−𝑐^∗₂2𝜑. (241) Then, 𝑣 satisfies (240). Thus, we conclude that −𝑐₂^∗2+4𝑚𝑎−𝑐^∗₂2𝜑≤𝜓𝜑≤−𝑐₂^∗2+4𝑚(1−𝜀)^𝑞(𝑎−𝜀)−𝑐^∗₂2𝜑. (242) When 𝑚<1, that is, 𝑝>1, (1−𝑚)𝑐₂^∗2+4𝑚(1−𝜀)^𝑞(𝑎−𝜀)−𝑐^∗₂2𝑚^1/(𝑚−1)≤lim_𝜉→+∞𝜑(𝜉)𝜉^1/(1−𝑚)≤(1−𝑚)𝑐₂^∗2+4𝑚𝑎−𝑐^∗₂2𝑚^1/(𝑚−1). (243) When 𝑚=1, that is, 𝑝=1, 𝑒^{−((𝑐2∗2+4𝑎−𝑐∗2)/2)𝜉}≤𝜑(𝜉)≤𝑒^{−((𝑐2∗2+4(1−𝜀)𝑞(𝑎−𝜀)−𝑐∗2)/2)𝜉}. (244) By the arbitrariness of 𝜀>0, (207)-(208) hold.

5. Discussion

When 𝑝=𝑞=1, the outcome in our work is reduced to the results obtained in [23]. Comparing with [23], our definition of sharp-type traveling wave fronts and smooth-type traveling wave fronts is more precise. In the proof of Lemma 3, for the case 𝑚+𝑝=2, we constructed two sequences to get the asymptotic expression of 𝜓(𝜑) for 𝜑>0 sufficiently small. This technique is not used in [23]. Moreover, the proof of Theorem 17 is more concise than the proof of Proposition 2.7 in [23].

Acknowledgments

The authors are grateful to the anonymous reviewers for the careful reading and helpful suggestions which led to an improvement of their original paper. This research is supported by the Fundamental Research Funds for the Central Universities (no. 2012ZM0057), the National Science Foundation of Guangdong Province (no. S2012040007959), and the National Natural Science Foundation of China (no. 11171115).