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

Authors: Weifang Yan; Rui Liu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

In this paper, we consider the traveling wave fronts for the following reaction diffusion equation with Hodgkin-Huxley source:(1)∂u∂t=∂2um∂x2+f(u,uτ),t≥0,x∈ℝ,
where m>0, f(u,v)=up(1-u)q(v-a), p>0, q>0, m+p>1, a∈(0,1) is a constant, and uτ(x,t)= u(x,t-τ) for τ>0.In 1952, Hodgkin and Huxley [1] proposed the Hodgkin-Huxley (H-H) equation
(2)dudt=u(t)(1-u(t))(u(t-τ)-a),a∈(0,1),
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:
(3)∂u∂t=∂2u∂x2+u(1-u)(uτ-a).
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:
(4)∂u∂t=∂2um∂x2+u(1-u)(u-a),a∈(0,1).
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:
(5)∂u∂t=∂∂x(D(u)∂u∂x)+u(1-u)(u-a),∂u∂t=∂∂x(D(u)∂u∂x)a∈(0,1),D(0)=0,
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:
(6)∂u∂t=∂2um∂x2+u(1-u)(uτ-a),∂u∂t=∂∂x(|∂u∂x|p-2∂u∂x)+uq(1-u)(uτ-a).
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:
(7)cφ′(ξ)=(φm)′′(ξ)+φ(ξ)p(1-φ(ξ))q(φcτ(ξ)-a),
where φcτ(ξ)=φ(ξ-cτ).Before going further, we first give the definition of sharp- and smooth-type traveling wave fronts.Definition 1.
A functionφ(ξ) is called a traveling wave front with wave speed c>0 if there exist ξa,ξb with -∞≤ξa<ξb≤+∞ such that φ∈C2(ξa,ξb) is monotonic increasing and
(8)cφ′(ξ)=(φm)′′(ξ)+φ(ξ)p(1-φ(ξ))q(φcτ(ξ)-a),cφ′(ξ)=(φm)′′(ξ)+φ(ξ)p(1-φ(ξ))qξ∈(ξa,ξb),(9)φ(ξ)=0forξa-cτ<ξ≤ξa,φ(ξb)=1,(φm)′(ξa)=(φm)′(ξb)=0,
or there exist ξ~a, ξ~b with -∞≤ξ~a<ξ~b≤+∞ such that φ∈C2(ξ~a,ξ~b) is monotonic decreasing and
(10)cφ′(ξ)=(φm)′′(ξ)+φ(ξ)p(1-φ(ξ))q(φcτ(ξ)-a),cφ′(ξ)=(φm)′′(ξ)+φ(ξ)p(1-φ(ξ))qξ∈(ξ~a,ξ~b),(11)φ(ξ)=1forξ~a-cτ<ξ≤ξ~a,φ(ξ~b)=0,(φm)′(ξ~a)=(φm)′(ξ~b)=0.(i)
Ifξa>-∞ and φ′(ξa+)≠0, then φ(ξ) is called an increasing sharp-type traveling wave front (see Figure 1(a)).(ii)
Ifξa=-∞ or φ′(ξa+)=0, then φ(ξ) is called an increasing smooth-type traveling wave front (see Figure 2(a)).Similarly, we have the following.(iii)
Ifξ~b<+∞ and φ′(ξ~b-)≠0, then φ(ξ) is called a decreasing sharp-type traveling wave front (see Figure 1(b)).(iv)
Ifξ~b=+∞ or φ′(ξ~b-)=0, then φ(ξ) is called a decreasing smooth-type traveling wave front (see Figure 2(b)).Sharp-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing.
(a)
(b)Smooth-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing.
(a)
(b)Letψ(ξ)=(φm)′(ξ). Then, (8) or (10) is transformed into
(12)φ′(ξ)=1mφ1-m(ξ)ψ(ξ),ψ′(ξ)=cmφ1-m(ξ)ψ(ξ)-φ(ξ)p(1-φ(ξ))q(φcτ(ξ)-a).
Furthermore, by (9) or (11), we give the asymptotic boundary conditions for traveling wave fronts as follows:
(13)φ(ξ)=0forξa-cτ<ξ≤ξa,φ(ξb)=1,ψ(ξa)=ψ(ξb)=0,
or
(14)φ(ξ)=1forξ~a-cτ<ξ≤ξ~a,φ(ξ~b)=0,ψ(ξ~a)=ψ(ξ~b)=0.
If ψ(ξ) is strictly positive or negative for 0<φ<1, then (12) is equivalent to
(15)dψdφ=c-mφm+p-1(1-φ)q(φcτ-a)ψ.
Clearly, for any given φ>0, if
(16)∫0φ‍mσm-1ψ(σ)dσ=+∞,
then φcτ can be defined by
(17)cτ=∫φcτφ‍mσm-1ψ(σ)dσ.
However, if for some φ>0,
(18)∫0φ‍mσm-1ψ(σ)dσ<+∞,
then ∫0φ‍(mσm-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 φcτ.(i) Ifψ is positive, define φcτ by
(19)cτ=∫φcτφ‍mσm-1ψ(σ)dσ,if∫0φ‍mσm-1ψ+(σ)dσ≥cτ,φcτ=0,if∫0φ‍mσm-1ψ+(σ)dσ<cτ,whereψ+ is a solution of the following problem:
(20)dψdφ=c+maφm+p-1(1-φ)qψ,forφ∈(0,1),ψ(0+)=0,ψ(φ)>0,forφ∈(0,1).
(ii) If ψ is negative, define φcτ by
(21)cτ=∫φφcτ‍mσm-1-ψ(σ)dσ,if∫φ1‍mσm-1-ψ-(σ)dσ≥cτ,φcτ=1,if∫φ1‍mσm-1-ψ-(σ)dσ<cτ,whereψ- is a solution of the following problem:
(22)dψdφ=c-m(1-a)φm+p-1(1-φ)qψ,forφ∈(0,1),ψ(1-)=0,ψ(φ)<0,forφ∈(0,1).
Consider the following problem:
(23)dψdφ=c-mφm+p-1(1-φ)q(φcτ-a)ψ,ψ(0+)=0,ψ(1-)=0.
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 solutionψ(φ) of the problem (23) with ψ(φ)>0 for φ∈(0,1).Sinceψ>0, we see that φ(ξ) is increasing in ξ, and so φcτ(ξ)≤φ(ξ). Thus, to investigate the behavior of the trajectories ψc(φ) of (23), we have to study the trajectories starting from (0,0), since the property of ψc at φ depends on the behavior of ψc at φcτ closely. Consider the following problem:
(24)dψdφ=c-mφm+p-1(1-φ)q(φcτ-a)ψforφ∈(0,β),ψ(0+)=0,ψ(φ)>0forφ∈(0,β),
where (0,β) with β≤1 is the maximal existence interval of the solution ψ(φ)>0. By (24),
(25)12ψ2(φ)=∫0φ‍cψ(s)ds-m∫0φ‍sm+p-1(1-s)q(scτ-a)ds≥∫0φ‍cψ(s)ds-m∫0φ‍sm+p-1(1-s)q(s-a)ds.
This excludes ψ(1-)=0 if ∫01‍sm+p-1(1-s)q(s-a)ds≤0. Thus, we only need to find the increasing traveling wave fronts for the case ∫01‍sm+p-1(1-s)q(s-a)ds>0. We first prove the following two lemmas.Lemma 2.
Assume that0<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<∫0φ0‍(mσm-1/ψ2(σ))dσ and φ1(ξ1-s)= φ2(ξ2-s)= 0 if s≥∫0φ0‍(mσm-1/ψ2(σ))dσ.Proof.
Recalling (19), we see that
(26)s=∫φisφ0‍mσm-1ψi(σ)dσif∫0φ0‍mσm-1ψi(σ)dσ>s,φis=0if∫0φ0‍mσm-1ψi(σ)dσ≤s,
where φis=φi(ξi-s),i=1,2. The desired conclusion follows immediately.Lemma 3.
For any givenτ>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).)The properties of the trajectoryψc(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.
(a)
(b)
(c)Proof.
We first show thatψ1(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 casem+p>2. According to (24), we have
(27)dψdφ=c+mφm+p-1(1-φ)q(a-φcτ)ψ.
Noticing that φcτ≤φ, we have for φ≤a that
(28)dψdφ≥c.
Integrating from 0 to φ yields
(29)ψ(φ)≥cφ.
We further have
(30)dψdφ=c+mφm+p-1(1-φ)q(a-φcτ)ψ≤c+mφm+p-1(1-φ)q(a-φcτ)cφ≤c+amcφm+p-2.
Integrating from 0 to φ gives
(31)ψ(φ)≤cφ+amc(m+p-1)φm+p-1.
That is,
(32)ψ(φ)~cφasφ⟶0+.
Therefore, ψ1(φ)>ψ2(φ) for sufficiently small φ>0.
(ii) Consider the casem+p=2. Similar to (i), we have
(33)ψ(φ)≥cφ,
and, hence,
(34)ψ(φ)≤(c+amc)φ.
Consider the sequences {An} and {Bn}, where
(35)A1=c,B1=c+amc,An=c+(1-ε)q(a-ε)mBn-1,Bn=c+amAn-1,n=2,3,…,
and ε>0 is sufficiently small. Noticing that A1<B1 and A1<A2, by induction, we obtain
(36)A1<A2=A3<A4=A5<⋯<An<⋯<Bn<⋯<B4=B3<B2=B1.
For 0<φ<ε, if
(37)An-1φ≤ψ(φ)≤Bn-1φ,
then
(38)dψdφ=c+mφ(1-φ)q(a-φcτ)ψ≤c+mφ(1-φ)q(a-φcτ)An-1φ≤c+amAn-1,dψdφ=c+mφ(1-φ)q(a-φcτ)ψ≥c+mφ(1-φ)q(a-φcτ)Bn-1φ≥c+(1-ε)q(a-ε)mBn-1.
Integrating from 0 to φ yields
(39)Anφ≤ψ(φ)≤Bnφ.
Thus, we have
(40)limn→∞An=A*(ε),limn→∞Bn=B*(ε),A*(ε)φ≤ψ(φ)≤B*(ε)φ,
with A*(ε) and B*(ε) satisfying
(41)A*(ε)=c+(1-ε)q(a-ε)mB*(ε),B*(ε)=c+amA*(ε).
Letting ε→0, we obtain
(42)limε→0A*(ε)=limε→0B*(ε)=c2+4ma+c2.
That is,
(43)ψ(φ)~c2+4ma+c2φasφ⟶0+.
Therefore, ψ1(φ)>ψ2(φ) for sufficiently small φ>0.
(iii) Consider the case1<m+p<2. Notice that
(44)dψdφ≥mφm+p-1(1-φ)q(a-φ)ψ,
which means that
(45)ψ2(φ)≥∫0φ‍2msm+p-1(1-s)q(a-s)ds=2mam+pφm+p+o(φm+p);
that is,
(46)ψ(φ)≥2mam+pφ(m+p)/2+o(φ(m+p)/2).
Consequently,
(47)dψdφ≤c+maφm+p-1ψ≤c+maφm+p-12ma/(m+p)φ(m+p)/2+o(φ(m+p)/2)=c+ma(m+p)2φ(m+p-2)/2+o(φ(m+p-2)/2).
That is,
(48)ψ(φ)≤cφ+2mam+pφ(m+p)/2+o(φ(m+p)/2).
Thus, we have
(49)ψ(φ)=2mam+pφ(m+p)/2+o(φ(m+p)/2).
Recalling (46), we see that
(50)dψ2dφ=2cψ+2mφm+p-1(1-φ)q(a-φcτ)≥2c2mam+pφ(m+p)/2+2mφm+p-1(1-φ)q(a-φ)+o(φ(m+p)/2),
which implies that
(51)ψ2(φ)≥4cm+p+22mam+pφ(m+p+2)/2+2mam+pφm+p+o(φ(m+p+2)/2).
On the other hand, by (49), we have
(52)dψ2dφ≤2cψ+2maφm+p-1=2c2mam+pφ(m+p)/2+2maφm+p-1+o(φ(m+p)/2),
and, hence,
(53)ψ2(φ)≤4cm+p+22mam+pφ(m+p+2)/2+2mam+pφm+p+o(φ(m+p+2)/2).
Summing up, we arrive at
(54)ψ2(φ)=4cm+p+22mam+pφ(m+p+2)/2+2mam+pφm+p+o(φ(m+p+2)/2),asφ⟶0+,
which implies that ψ1(φ)>ψ2(φ) for sufficiently small φ>0.
We claim thatψ1(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,
(55)ψ1′(φ0)≤ψ2′(φ0),
which means that
(56)c1+mφ0m+p-1(1-φ0)q(a-φ0c1τ)ψ0≤c2+mφ0m+p-1(1-φ0)q(a-φ0c2τ)ψ0;
that is,
(57)0<c1-c2≤mφ0m+p-1(1-φ0)qψ0(φ0c1τ-φ0c2τ).
Thus,
(58)φ0c1τ>φ0c2τ.
Denote that φ1(ξ1)=φ2(ξ2)=φ0. By Lemma 2, we have
(59)φ1(ξ1-c1τ)≤φ2(ξ2-c1τ)≤φ2(ξ2-c2τ),
which means that φ0c1τ≤φ0c2τ, a contradiction.
In what follows, we will show thatψ1(β2)>0. Recalling the first equation of (24), we infer that
(60)12dψi2dφ=ciψi-mφm+p-1(1-φ)q(φciτ-a),i=1,2.
By Lemma 2, for any φ1(ξ1)=φ2(ξ2)<β2, we have
(61)φ1(ξ1-c1τ)≤φ2(ξ2-c1τ)≤φ2(ξ2-c2τ),
since ψ1>ψ2. Thus, we obtain
(62)dψ12dφ>dψ22dφ
for any φ∈(0,β2). Integrating the previous inequality from φ0 to φ for any 0<φ0<φ<β2 yields
(63)ψ12(φ)>ψ22(φ)+ψ12(φ0)-ψ22(φ0).
Letting φ→β2 gives
(64)ψ12(β2)≥ψ12(φ0)-ψ22(φ0)>0;
namely, ψ1(β2)>0. The proof is completed.To deal with the behavior of the trajectoriesψc(φ) of the problem (24), we introduce the level set
(65)Lk={(φ,ψ)∈ℝ2;12ψ2+m∫0φ‍sm+p-1(1-s)q(s-a)ds=k}
for any k≥0. Clearly, for c=0, the level sets {Lk} are exactly the trajectories of solutions to (12) or (15). Now, we define
(66)Lk+=Lk∩{ψ>0}.
Notice that if (φ,ψ)∈Lk+ is a solution of the system (12), we have
(67)dk(ξ)dξ=ψψ′+mφm+p-1(1-φ)q(φ-a)φ′=cmφ1-mψ2+ψφp(1-φ)q(φ-φcτ)≥cmφ1-mψ2>0,
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
(68)k1=m∫01‍sm+p-1(1-s)q(s-a)ds;
we know that Lk1 passes through the critical point (1,0). Denote that
(69)ψ*(φ)=2m∫φ1‍sm+p-1(1-s)q(s-a)ds.
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 trajectoryψc(φ) of the problem (24) must intersect with  ψ*(φ) for sufficiently large c>0. (See Figure 3(c).)Proof.
For anyφ∈(0,a], we have φcτ∈(0,a], and
(70)ψc′(φ)≥c>0.
Let φ0∈(a,1) be the first point such that ψc′(φ0)=0. Then, we have
(71)ψc(φ0)=mφ0m+p-1(1-φ0)q(φ0cτ-a)c.
Since
(72)ψc(φ0)>ψc(a)≥ac,
we have
(73)mφ0m+p-1(1-φ0)q(φ0cτ-a)>ac2.
Thus,
(74)mφ0m+p-1(1-φ0)q(φ0-a)>ac2.
Denote that
(75)M=maxφ∈(a,1){mφm+p-1(1-φ)q(φ-a)a}.
Then, for any c≥M, (74) does not hold and ψc(φ) is increasing on (a,1). Therefore, ψc(φ) must intersect with ψ*(φ) for any c≥M. The proof is completed.Theorem 5.
(i) If∫01‍sm+p-1(1-s)q(s-a)ds≤0, then there is no nontrivial nonnegative solution for problem (23).
(ii) If∫01‍sm+p-1(1-s)q(s-a)ds>0, then there exists a unique c1*>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∫01‍sm+p-1(1-s)q(s-a)ds≤0 has been discussed.
(ii) We know that for any fixedc>0, (φ,ψc) wanders through increasing level sets {Lk} strictly. Thus, (φ,ψc) intersects with a level set Lk at most once. Let (σc,ϕc) be the intersection point of (φ,ψc) with Lk1. Then, if ϕc>0, we have
(76)dψ*dφ|φ=σc=-mφm+p-1(1-φ)q(φ-a)ψ*|φ=σc≤-mφm+p-1(1-φ)q(φcτ-a)ϕc|φ=σc=dψcdφ|φ=σc-c.
Define
(77)c1*=inf{c>0;ψcintersectsψ*at(σc,ϕc)withσc∈(0,1]}.
By Lemma 4, c1* is well defined. In what follows, we will show that c1* is just the desired wave speed.
We first havec1*>0. Indeed, when c=0, the first equation of (24) becomes
(78)dψ0dφ=-mφm+p-1(1-φ)q(φ-a)ψ0.
Noticing that ψ0(0+)=0, we have
(79)ψ02(φ)=-2m∫0φ‍sm+p-1(1-s)q(s-a)ds.
Then, there exists φ0∈(a,1) such that
(80)ψ0(φ0)=0,
since ∫01‍sm+p-1(1-s)q(s-a)ds>0. The continuous dependence of ψc on c ensures that ψc goes to zero before reaching ψ* for sufficiently small c>0. Thus, c1*>0.
In addition, by Lemma3, we know that σc is decreasing on c. Assume that
(81)limc↘c1*σc=σc1*.
If σc1*<1; namely, ϕc1*>0, then by (76), we arrive at
(82)ψc1*′(σc1*)≥ψ*′(σc1*)+c1*>ψ*′(σc1*).
So, there exists σc1*<φ1<1, such that ψc1*(φ1)>ψ*(φ1). By the continuous dependence of ψc on c, there is c>0 with c<c1* and close to c1* sufficiently such that ψc(φ1)≥ψ*(φ1), which implies that ψc intersects with  ψ*. This contradicts the definition of c1*. Thus, σc1*=1, which implies ϕc1*=0.
Moreover, by Lemma3, for c>c1*, we have ψc(1)>0. However, if 0<c<c1*, there exists φc with a<φc<1 such that ψc(φ)→0 as φ↗φc. 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. Letψ(φ)>0 for any φ∈(0,1) be a solution of the problem (23), and φ(ξ) solves
(83)φ′(ξ)=1mφ1-m(ξ)ψ(φ(ξ)).
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
(84)(φm)′′(ξ)=ψ′(φ)φ′(ξ)=(c-mφm+p-1(1-φ)q(φcτ-a)ψ)φ′=cφ′-φp(1-φ)q(φcτ-a).
Moreover,
(85)φ(ξ)=0forα-cτ<ξ≤α,φ(β)=1,(φm)′(α)=(φm)′(β)=0.
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.Theorem5 and Proposition 6 imply the following result.Theorem 7.
(i) If∫01‍sm+p-1(1-s)q(s-a)ds≤0, then there is no increasing traveling wave front for the problem (8)-(9).
(ii) If∫01‍sm+p-1(1-s)q(s-a)ds>0, then there is a unique wave speed c1*>0, such that the problem (8)-(9) admits an increasing traveling wave front.Furthermore, we have the following results.Theorem 8.
Letφ(ξ) be the traveling wave front of the problem (8)-(9) corresponding to the wave speed c1*.(i)
Ifm<2, then φ(ξ) is a smooth-type traveling wave front.(ii)
Ifm=2, then φ′(ξa+)=c1*/2 and φ(ξ) is a sharp-type traveling wave front.(iii)
Ifm>2, then φ′(ξa+)=+∞ and φ(ξ) is a sharp-type traveling wave front.Proof.
(i) If0<m≤1, it is easy to see that
(86)φ′(ξa+)=1mφ1-m(ξa+)ψ(ξa+)=0.
If 1<m<2, from the proof of Lemma 3, we see that when φ>0 is sufficiently small,
(87)ψ(φ)=c1*φ+o(φ),m+p>2;ψ(φ)=c1*2+4ma+c1*2φ+o(φ),m+p=2;ψ(φ)=2mam+pφ(m+p)/2+o(φ(m+p)/2),1<m+p<2.
Since
(88)φ′(ξ)=1mφ1-m(ξ)ψ(ξ),
we have
(89)φ′(ξ)=c1*mφ2-m(ξ)+o(φ2-m(ξ)),m+p>2;φ′(ξ)=c1*2+4ma+c1*2mφ2-m(ξ)+o(φ2-m(ξ)),φ′(ξ)=c1*2+4ma+c1*2mφ2-m(ξ)m+p=2;φ′(ξ)=2am(m+p)φ(2-m+p)/2(ξ)+o(φ(2-m+p)/2(ξ)),φ′(ξ)=c1*2+4ma+c1*2mφ2-m(ξ)1<m+p<2.
Thus,
(90)φ′(ξa+)=0.
(ii) Ifm=2, we have
(91)φ′(ξa+)=c1*2.
(iii) Ifm>2, we have
(92)φ′(ξa+)=+∞.
The proof is completed.Theorem 9.
c
1
*
(
τ
) is nonincreasing in delay τ; namely, if τ1>τ2, then c1*(τ1)≤c1*(τ2).Proof.
Suppose for contradiction that there exist time delaysτ1 and τ2, with τ1>τ2 and c1*(τ1)>c1*(τ2). Denote that c^1=c1*(τ1) and  c^2=c1*(τ2) for simplicity. Take c such that
(93)c^2<c<c^1.
In what follows, denote the solutions of (24) corresponding to wave speed and time delay (c^1,τ1), (c,τ1), (c^2,τ2) by ψc^1, ψc, ψc^2, respectively.
Similar to the proof of Lemma3, we know that when φ>0 is sufficiently small, for m+p>2,
(94)ψc(φ)=cφ+o(φ),ψc^2(φ)=c^2φ+o(φ);
for m+p=2,
(95)ψc(φ)=c2+4ma+c2φ+o(φ),ψc^2(φ)=c^22+4ma+c^22φ+o(φ);
for 1<m+p<2,
(96)ψc2(φ)=4cm+p+22mam+pφ(m+p+2)/2+2mam+pφm+p+o(φ(m+p+2)/2),ψc^22(φ)=4c^2m+p+22mam+pφ(m+p+2)/2+2mam+pφm+p+o(φ(m+p+2)/2).
Since c>c^2, we have
(97)ψc(φ)>ψc^2(φ)
for sufficiently small φ>0. Furthermore, we claim that ψc(φ)>ψc^2(φ) for φ∈(0,1). Otherwise, there exists φ0∈(0,1) such that ψc(φ)>ψc^2(φ) for φ∈(0,φ0), and ψc(φ0)=ψc^2(φ0)=ψ0. Then, we have
(98)ψc′(φ0)≤ψc^2′(φ0);
that is,
(99)0<c-c^2≤mφ0m+p-1(1-φ0)q(φ0cτ1-φ0c^2τ2)ψ0,
which implies φ0cτ1>φ0c^2τ2. On the other hand, by Lemma 2, we have φ0cτ1≤φ0c^2τ2, a contradiction. Similarly, we can get ψc^1(φ)>ψc(φ). From the uniqueness of wave speed on any fixed τ>0, this contradicts the definition of c^1.


## 3. Existence of Decreasing Traveling Waves

In this section, we aim to find a solutionψ(φ) of the problem (23) with ψ(φ)<0 for φ∈(0,1). We first introduce a comparison lemma.Lemma 10.
Letu,v be the solutions of the following problems, respectively:
(100)dudφ=c-mφm+p-1(1-φ)q(φ-a)uforφ<1,u(1-)=0,dvdφ=c-mφm+p-1(1-φ)q(1-a)vforφ<1,v(1-)=0.
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 thatφcτ≤φ when ψ is positive, and thus
(101)12dψ2dφ-cψ=-mφm+p-1(1-φ)q(φcτ-a)≥-mφm+p-1(1-φ)q(φ-a)=12du2dφ-cu;
that is,
(102)d(ψ2-u2)dφ-2c(ψ2-u2)g(φ)≥0,
where
(103)g(φ)=1ψ+u.
Consequently,
(104)d((ψ2-u2)e2c∫φ1‍g(s)ds)dφ≥0.
Integrating from φ to 1 yields
(105)ψ2(φ)-u2(φ)≤0,
and so ψ(φ)≤u(φ) if ψ,u>0. Similarly, u(φ)≤v(φ) if u,v>0.
The proof forψ,u,v<0 is similar and omitted here.Sinceψ<0, we see that φ(ξ) is decreasing in ξ, and so φcτ(ξ)≥φ(ξ). To get the behavior of the trajectories ψc(φ) of (23), we have to study the trajectories starting from (1,0), since the property of ψc at φ depends on the behavior of ψc at φcτ closely. Consider the following problem:
(106)dψdφ=c-mφm+p-1(1-φ)q(φcτ-a)ψforφ∈(α,1),ψ(1-)=0,ψ(φ)<0forφ∈(α,1),
where (α,1) with α≥0 is the maximal existence interval of the solution ψ(φ)<0. By (106),
(107)12ψ2(φ)=-∫φ1‍cψ(s)ds+m∫φ1‍sm+p-1(1-s)q(scτ-a)ds≥-∫φ1‍cψ(s)ds+m∫φ1‍sm+p-1(1-s)q(s-a)ds.
Hence, ψ(0+)=0 is impossible if ∫01‍sm+p-1(1-s)q(s-a)ds≥0. So, we only need to find the decreasing traveling wave fronts for the case ∫01‍sm+p-1(1-s)q(s-a)ds<0. We first give the following three lemmas.Lemma 11.
Assume that0<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<∫φ01‍(mσm-1/-ψ2(σ))dσ, and φ1(ξ1-s)= φ2(ξ2-s)=1 if s≥∫φ01(mσm-1/-ψ2(σ)‍)dσ.Proof.
The proof is similar to that of Lemma2.Lemma 12.
For any givenτ>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).)The properties of the trajectoryψc(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.
(a)
(b)
(c)Proof.
We first show thatψ1(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 caseq>1. According to (106), we have
(108)dψdφ=c-mφm+p-1(1-φ)q(φcτ-a)ψ.
Noticing that φcτ≥φ, we have for φ≥a that
(109)dψdφ≥c.
Integrating from φ to 1 yields
(110)-ψ(φ)≥c(1-φ).
We further have
(111)dψdφ=c-mφm+p-1(1-φ)q(φcτ-a)ψ≤c+mφm+p-1(1-φ)q(φcτ-a)c(1-φ)≤c+m(1-a)c(1-φ)q-1.
Integrating from φ to 1 gives
(112)-ψ(φ)≤c(1-φ)+m(1-a)cq(1-φ)q.
Thus,
(113)ψ(φ)~c(φ-1),asφ⟶1-.
Therefore, ψ1(φ)<ψ2(φ) in a left neighborhood of φ=1.
(ii)   Whenq=1, consider the following two systems:
(114)φ′(ξ)=1mφ1-m(ξ)u1(ξ),u1′(ξ)=cmφ1-m(ξ)u1(ξ)-φ(ξ)p(1-φ(ξ))(φ(ξ)-a),φ′(ξ)=1mφ1-m(ξ)u2(ξ),u2′(ξ)=cmφ1-m(ξ)u2(ξ)-φ(ξ)p(1-φ(ξ))(1-a).
Notice that the right hand side of the previous two systems shares the same Jacobian matrix
(115)P=(01m1-acm)
at (1,0). By a simple calculation, we get the eigenvalues of the matrix P as
(116)λ+=c+c2+4m(1-a)2m,λ-=c-c2+4m(1-a)2m.
It is easy to see that (1,0) is a saddle point. And the eigenvector associated with the eigenvalue λ+ can be
(117)Q=(1mλ+).
We express the local explicit solutions of the problem (114) in the left neighborhood of (1,0) to reach
(118)φ(ξ)=1-leλ+ξ+𝒪((1-φ)2),ui(ξ)=-lmλ+eλ+ξ+𝒪(ui2),i=1,2.
Therefore, near (1,0), we have
(119)ui~mλ+(φ-1)=c+c2+4m(1-a)2(φ-1),i=1,2.
By the comparison lemma, u2≤ψ≤u1, which implies
(120)ψ(φ)~c+c2+4m(1-a)2(φ-1),asφ⟶1-.
Thus, ψ1(φ)<ψ2(φ) in a left neighborhood of φ=1.
(iii) Consider the case0<q<1. Notice that
(121)dψdφ≥mφm+p-1(1-φ)q(a-φ)ψ,
which means that
(122)-ψ2(φ)≤∫φ1‍2msm+p-1(1-s)q(a-s)ds=2m(a-1)q+1(1-φ)q+1+o((1-φ)q+1);
that is,
(123)-ψ(φ)≥2m(1-a)q+1(1-φ)(q+1)/2+o((1-φ)(q+1)/2).
Consequently,
(124)dψdφ≤c-m(1-a)(1-φ)qψ≤c+m(1-a)(1-φ)q2m(1-a)/(q+1)(1-φ)(q+1)/2+o((1-φ)(q+1)/2)=c+m(1-a)(q+1)2(1-φ)(q-1)/2+o((1-φ)(q-1)/2).
That is,
(125)-ψ(φ)≤c(1-φ)+2m(1-a)q+1(1-φ)(q+1)/2+o((1-φ)(q+1)/2).
Thus, we have
(126)-ψ(φ)=2m(1-a)q+1(1-φ)(q+1)/2+o((1-φ)(q+1)/2).
Recalling (123), we see that
(127)-dψ2dφ=-2cψ+2mφm+p-1(1-φ)q(φcτ-a)≥2c2m(1-a)q+1(1-φ)(q+1)/2+2mφm+p-1(1-φ)q(φ-a)+o((1-φ)(q+1)/2),
which implies that
(128)ψ2(φ)≥4cq+32m(1-a)q+1(1-φ)(q+3)/2+2m(1-a)q+1(1-φ)q+1+o((1-φ)q+1).
On the other hand, by (126), we have
(129)-dψ2dφ≤-2cψ+2m(1-a)(1-φ)q=2c2m(1-a)q+1(1-φ)(q+1)/2+2m(1-a)(1-φ)q+o((1-φ)(q+1)/2),
and, hence,
(130)ψ2(φ)≤4cq+32m(1-a)q+1(1-φ)(q+3)/2+2m(1-a)q+1(1-φ)q+1+o((1-φ)q+1).
Summing up, we arrive at
(131)ψ2(φ)=4cq+32m(1-a)q+1(1-φ)(q+3)/2+2m(1-a)q+1(1-φ)q+1+o((1-φ)q+1),&+2m(1-a)q+1(1-φ)q+1asφ⟶1-,
which implies that ψ1(φ)<ψ2(φ) in a left neighborhood of φ=1.
The proof for the claims thatψ1(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 Section2, we denote level set Rk by
(132)Rk={(φ,ψ)∈ℝ2;12ψ2-m∫φ1‍sm+p-1(1-s)q(s-a)ds=k}
for any k≥0, and correspondingly, define
(133)Rk-=Rk∩{ψ<0}.
Notice that if (φ,ψ)∈Rk- solves system (12), then
(134)dk(ξ)dξ=ψψ′+mφm+p-1(1-φ)q(φ-a)φ′=cmφ1-mψ2+ψφp(1-φ)q(φ-φcτ)≥cmφ1-mψ2>0,
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
(135)k2=-m∫01‍sm+p-1(1-s)q(s-a)ds,
we know that Rk2 passes through the critical point (0,0). Denote that
(136)ψ~*(φ)=--2m∫0φ‍sm+p-1(1-s)q(s-a)ds.
We introduce the following lemma.Lemma 13.
The trajectoryψc(φ) of the problem (106) must intersect with  ψ~*(φ) for sufficiently large c>0. (See Figure 4(c).)Proof.
For anyφ∈[a,1), we have φcτ∈[a,1) and
(137)ψc′(φ)≥c>0.
Let φ0∈(0,a) be the first point such that ψc′(φ0)=0. Then, we have
(138)ψc(φ0)=mφ0m+p-1(1-φ0)q(φ0cτ-a)c.
Since
(139)ψc(φ0)<ψc(a)≤(a-1)c,
we have
(140)mφ0m+p-1(1-φ0)q(φ0cτ-a)<(a-1)c2.
Thus,
(141)mφ0m+p-1(1-φ0)q(φ0-a)<(a-1)c2.
Denote that
(142)M~=maxφ∈(0,a){mφm+p-1(1-φ)q(a-φ)1-a}.
Then, for any c≥M~, (141) does not hold, and ψc(φ) is increasing on (0,a). Therefore, ψc(φ) must intersect with  ψ~*(φ) for any c≥M~. The proof is completed.Theorem 14.
(i) If∫01‍sm+p-1(1-s)q(s-a)ds≥0, then there is no nontrivial nonpositive solution for the problem (23).
(ii) If∫01‍sm+p-1(1-s)q(s-a)ds<0, then there exists a unique c2*>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 Theorem5, and
(143)c2*=inf{c>0;ψcintersectsψ~*at(ηc,ϕ~c)withηc∈[0,1)}.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 Proposition6.Theorem14 and Proposition 15 imply the following result.Theorem 16.
(i) If∫01‍sm+p-1(1-s)q(s-a)ds≥0, then there is no decreasing traveling wave front for the problem (10)-(11).
(ii) If∫01‍sm+p-1(1-s)q(s-a)ds<0, then there is a unique wave speed c2*>0, such that the problem (10)-(11) admits a decreasing traveling wave front.Furthermore, we have the following results.Theorem 17.
The traveling wave frontφ(ξ) of the problem (10)-(11) corresponding to the wave speed c2* obtained earlier is of smooth type.Proof.
It is easy to see that(144)dψdφ≤0,asφ⟶0+.
Thus,
(145)ψ≥mφm+p-1(1-φ)q(φc2*τ-a)c2*≥-mac2*φm+p-1.
Recalling that
(146)φ′(ξ)=1mφ1-m(ξ)ψ(ξ),
we know that
(147)-ac2*φp(ξ)≤φ′(ξ)≤0,
which means that
(148)φ′(ξ~b-)=0.
The proof is completed.Theorem 18.
c
2
*
(
τ
) is nonincreasing in delay τ; namely, if τ1>τ2, then c2*(τ1)≤c2*(τ2).Proof.
The proof is similar to that of Theorem9.


## 4. Asymptotic Behavior of the Traveling Wave Solutions

In this section, we first pay our attention to the finiteness ofξa,ξb,ξ~a,ξ~b. On the basis of this, the convergent rates of φ going to the steady states at far-field are discussed.Theorem 19.
Letφ(ξ) be the increasing traveling wave solution of the problem (8)-(9) corresponding to the unique wave speed c1*. (i)
If0<q<1, then ξb<+∞.(ii)
Ifq≥1, then ξb=+∞. More precisely,
(a)
whenq>1, one has
(149)1-φ(ξ)~((1-a)(q-1)c1*ξ)1/(1-q),asξ⟶+∞;(b)
whenq=1, one has
(150)1-φ(ξ)~e-((c1*2+4m(1-a)-c1*)/2m)ξ,asξ⟶+∞.Proof.
(i) For(1+a)/2≤φc1*τ<1, we have (1+a)/2≤φc1*τ≤φ≤1, and
(151)12dψ2dφ-c1*ψ=-mφm+p-1(1-φ)q(φc1*τ-a)≤-m(1-a)2(1+a2)m+p-1(1-φ)q.
Consider the following inequality problem:
(152)12du2dφ-c1*u≥-m(1-a)2(1+a2)m+p-1(1-φ)q,12du2dφ-c1*u≥-m(1-a)2(1+a2)m+p-1forφ≤1,u(1-)=0,u(φ)>0,forφ<1.
It is easy to prove that ψ(φ)≥u(φ) when φc1*τ≥(1+a)/2. Now, we construct a function u satisfying (152). Let
(153)u=A(1-φ)θ.
Then, u satisfies (152) if and only if
(154)A2θ(1-φ)2θ-1-q+c1*A(1-φ)θ-q≤m(1-a)2(1+a2)m+p-1.
Take
(155)θ=max{1+q2,q}.
Then, (154) holds when A is appropriately small. Thus,
(156)ψ(φ)≥A(1-φ)θ,
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,
(157)ξ(φ)-ξ(φ0)=∫φ0φ‍msm-1ψ(s)ds≤mA∫φ0φ‍sm-1(1-s)θds.
When 0<q<1,ξ(φ)<+∞ as φ→1-. Thus, ξb<+∞.
(ii) It is easy to see that(158)dψdφ≤0,asφ⟶1-.
Thus,
(159)ψ≤mφm+p-1(1-φ)q(φc1*τ-a)c1*≤mφm+p-1(1-φ)q(φ-a)c1*,ξ(φ)-ξ(φ0)=∫φ0φ‍msm-1ψ(s)ds≥∫φ0φ‍c1*sp(1-s)q(s-a)ds.
Letting φ→1, we obtain
(160)limφ→1-ξ(φ)=+∞;
that is,
(161)ξb=+∞.
For anyε>0, when φ approaches 1 enough, we have φc1*τ>1-ε. Consider the following two problems:
(162)12du2dφ-c1*u≤-mφm+p-1(1-φ)q(1-a),12du2dφ-c1*u≤-mφm+p-1(1-φ)qforφ≤1,u(1-)=0,u(φ)>0forφ<1,(163)12dv2dφ-c1*v≥-mφm+p-1(1-φ)q(1-ε-a),12dv2dφ-c1*v≥-mφm+p-1(1-φ)qforφ≤1,v(1-)=0,v(φ)>0forφ<1.
It is easy to prove that v(φ)≤ψ(φ)≤u(φ) when φ<1 with φc1*τ>1-ε. In what follows, we construct two functions u and v satisfying (162) and (163), respectively. Let
(164)u=A(1-φ)θ.
Then, u satisfies (162) if and only if
(165)A2θ(1-φ)2θ-1-q+c1*A(1-φ)θ-q≥m(1-a)φm+p-1.
When q>1, take
(166)θ=q,A=m(1-a)c1*.
When q=1, take
(167)θ=1,A=c1*2+4m(1-a)-c1*2.
Then, u is a solution of problem (162).
On the other hand, let(168)v=A~(1-φ)θ~.
Then, v satisfies (163) if and only if
(169)A~2θ~(1-φ)2θ~-1-q+c1*A~(1-φ)θ~-q≤m(1-ε-a)φm+p-1.
When q>1, take θ~=q; then, (169) is ensured by
(170)A~2qεq-1+c1*A~≤m(1-ε-a)(1-ε)m+p-1(1-ε(q-1)/2+ε(q-1)/2),
which holds if
(171)A~2qε(q-1)/2≤m(1-ε-a)(1-ε)m+p-1,c1*A~≤m(1-ε-a)(1-ε)m+p-1(1-ε(q-1)/2).
Take
(172)A~=m(1-ε-a)(1-ε)m+p-1(1-ε(q-1)/2)c1*.
Then, when
(173)ε≤min{1-a,c1*4/(q-1)(m(1-a))2/(q-1)q2/(q-1)},
(169) holds. When q=1, take θ~=1, and
(174)A~=c1*2+4m(1-ε-a)(1-ε)m+p-1-c1*2.
Then, (169) is ensured.
Summing up, for anyφ<1 with φc1*τ>1-ε, when q>1,
(175)m(1-ε-a)(1-ε)m+p-1(1-ε(q-1)/2)c1*(1-φ)q≤ψ(φ)≤m(1-a)c1*(1-φ)q.
Noticing that
(176)ξ(φ)=ξ(φ0)+∫φ0φ‍msm-1ψ(s)ds,
we have
(177)c1*(1-a)(q-1)≤limφ→1-ξ(φ)(1-φ)q-1≤c1*(1-ε-a)(1-ε)m+p-1(1-ε(q-1)/2)(q-1);
that is,
(178)((1-a)(q-1)c1*)1/(1-q)≤limξ→+∞(1-φ(ξ))ξ1/(q-1)≤((1-ε-a)(1-ε)m+p-1(1-ε(q-1)/2)(q-1)c1*)1/(1-q).
When q=1,
(179)c1*2+4m(1-ε-a)(1-ε)m+p-1-c1*2(1-φ)≤ψ(φ)≤c1*2+4m(1-a)-c1*2(1-φ).
A direct calculation gives
(180)-2mc1*2+4m(1-a)-c1*≤limφ→1-ξ(φ)ln(1-φ)≤-2mc1*2+4m(1-ε-a)(1-ε)m+p-1-c1*;
that is,
(181)e-((c1*2+4m(1-a)-c1*)/2m)ξ≤1-φ(ξ)≤e-((c1*2+4m(1-ε-a)(1-ε)m+p-1-c1*)/2m)ξ.
By the arbitrariness of ε>0, (149)-(150) hold.Theorem 20.
Letφ(ξ) be the increasing traveling wave solution of the problem (8)-(9) corresponding to the unique wave speed c1*. (i)
Ifm>min{p,1}, then ξa>-∞.(ii)
Ifm≤min{p,1}, then ξa=-∞. More precisely,
(a)
whenm+p>2, if m=1, one has
(182)φ(ξ)~ec1*ξ,asξ⟶-∞,and ifm<1, one has
(183)φ(ξ)~((m-1)c1*mξ)1/(m-1),asξ⟶-∞;(b)
whenm+p=2, if m=1, one has
(184)φ(ξ)~e((c1*2+4a+c1*)/2)ξ,asξ⟶-∞,and ifm<1, one has
(185)φ(ξ)~((m-1)(c1*2+4ma+c1*)2mξ)1/(m-1),((m-1)c1*mξ)1/(m-1)asξ⟶-∞;(c)
when1<m+p<2, noticing that m≤min{p,1}, then m≤p; if m=p, one has
(186)φ(ξ)~e(a/m)ξ,asξ⟶-∞,and ifm<p, one has
(187)φ(ξ)~(m-p2m2mam+pξ)2/(m-p),asξ⟶-∞.Proof.
For anyφ0∈(0,a), we see that
(188)ξ(φ)=ξ(φ0)-∫φφ0‍msm-1ψ(s)ds.
From the proof of Lemma 3, we see that when φ>0 is sufficiently small,
(189)ψ(φ)=Aφγ+o(φγ),
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 ξa>-∞; however, if m≤min{p,1}, then ξa=-∞. Furthermore, notice that as φ→0+, when m+p>2,
(190)ψ(φ)=c1*φ+o(φ).
If m=1, we have
(191)limφ→0+ξ(φ)lnφ=1c1*.
If m<1,
(192)limφ→0+ξ(φ)φm-1=m(m-1)c1*.
When m+p=2,
(193)ψ(φ)=c1*2+4ma+c2φ+o(φ).
If m=1, we have
(194)limφ→0+ξ(φ)lnφ=c1*2+4a-c1*2a.
If m<1,
(195)limφ→0+ξ(φ)φm-1=c1*2+4ma-c1*2a(m-1).
When 1<m+p<2,
(196)ψ(φ)=2mam+pφ(m+p)/2+o(φ(m+p)/2).
If m=p, we have
(197)limφ→0+ξ(φ)lnφ=ma.
If m<p,
(198)limφ→0+ξ(φ)φ(m-p)/2=2mm-pm+p2ma.
By a simple calculation, (182)–(187) hold.Theorem 21.
Letφ(ξ) be the decreasing traveling wave solution of the problem (10)-(11) corresponding to the unique wave speed c2*. (i)
If0<q<1, then ξ~a>-∞.(ii)
Ifq≥1, then ξ~a=-∞. More precisely,
(a)
whenq>1, one has
(199)1-φ(ξ)~e(c2*/m)ξ,asξ⟶-∞;(b)
whenq=1, one has
(200)1-φ(ξ)~e((c2*2+4m(1-a)+c2*)/2m)ξ,asξ⟶-∞.Proof.
Letφ0 approach 1 enough. Then, we have
(201)ξ(φ0)-ξ(φ)=∫φφ0‍msm-1ψ(s)ds.
From the proof of Lemma 12, we see that
(202)ψ(φ)~A(1-φ)γ,
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 ξ~a>-∞; however, if q≥1, then ξ~a=-∞. Furthermore, notice that as φ→1-, when q>1,
(203)ψ(φ)=c2*(φ-1)+o(φ-1),limφ→1-ξ(φ)ln(1-φ)=mc2*.
When q=1,
(204)ψ(φ)=c2*2+4m(1-a)+c2*2(φ-1)+o(φ-1),limφ→1-ξ(φ)ln(1-φ)=2mc2*2+4m(1-a)+c2*.
This yields (199)-(200).Theorem 22.
Letφ(ξ) be the decreasing traveling wave solution of the problem (10)-(11) corresponding to the unique wave speed c2*. (i)
Ifp<min{m,1}, then ξ~b<+∞.(ii)
Ifp≥min{m,1}, then ξ~b=+∞. More precisely,
(a)
whenm+p>2, if p>1, one has
(205)φ(ξ)~(a(p-1)c2*ξ)1/(1-p),asξ⟶+∞,and ifp=1, one has
(206)φ(ξ)~e-(a/c2*)ξ,asξ⟶+∞;(b)
whenm+p=2, if p>1, one has
(207)φ(ξ)~((1-m)(c2*2+4ma-c2*)2mξ)1/(m-1),(a(p-1)c2*ξ)1/(1-p)asξ⟶+∞,and ifp=1, one has
(208)φ(ξ)~e-((c2*2+4a-c2*)/2)ξ,asξ⟶+∞;(c)
when1<m+p<2, noticing that p≥min{m,1}, then p≥m; if p>m, one has
(209)φ(ξ)~(p-m2m2mam+pξ)2/(m-p),asξ⟶+∞,and ifp=m, one has
(210)φ(ξ)~e-(a/m)ξ,asξ⟶+∞.Proof.
(i) For0<φc2*τ≤a/2, we have 0≤φ≤φc2*τ≤a/2, and
(211)12dψ2dφ-c2*ψ=-mφm+p-1(1-φ)q(φc2*τ-a)≥ma2(1-a2)qφm+p-1.
Consider the following inequality problem:
(212)12du2dφ-c2*u≤ma2(1-a2)qφm+p-1,forφ≥0,u(0+)=0,u(φ)<0,forφ>0.
It is easy to prove that ψ(φ)≤u(φ) when φc2*τ≤a/2. Now, we construct a function u satisfying (212). Let
(213)u=-Bφr.
Then, u satisfies (212) if and only if
(214)B2rφ2r-m-p+c2*Bφr-m-p+1≤ma2(1-a2)q.
Take
(215)r=max{m+p2,m+p-1}.
Then, (212) holds when B is appropriately small. Thus,
(216)ψ(φ)≤-Bφr,
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
(217)ξ(φ)-ξ(φ0)=-∫φφ0‍msm-1ψ(s)ds≤mB∫φφ0‍1sθ-m+1ds.
When p<min{m,1}, ξ(φ)<+∞ as φ→0+. Thus, ξ~b<+∞.
(ii) From the proof of Theorem14, we know that
(218)ψ(φ)≥ψ~*(φ)=-2m∫0φ‍sm+p-1(1-s)q(a-s)ds≥-2mam+pφ(m+p)/2.
On the other hand, from the proof of Theorem 17, we also note that
(219)ψ(φ)≥-mac2*φm+p-1.
By (218)-(219) and
(220)ξ(φ)-ξ(φ0)=∫φ0φ‍msm-1ψ(s)ds,
it is easy to see that
(221)limφ→0+ξ(φ)=+∞,
if p≥min{m,1}. That is,
(222)ξ~b=+∞.
For any0<ε<a, when φ approaches 0 enough, we have φc2*τ<ε. Consider the following problem:
(223)12du2dφ-c2*u≤mφm+p-1(1-φ)q(a-ε),forφ≥0,u(0+)=0,u(φ)<0,forφ>0.
It is easy to prove that ψ(φ)≤u(φ) when φ>0 with φc2*τ<ε. In what follows, we construct a function u satisfying (223). Let
(224)u=-Bφr.
Noticing that 0<φ≤φc2*τ<ε, then u satisfies (223) if
(225)B2rφ2r-m-p+c2*Bφr-m-p+1≤m(1-ε)q(a-ε).
When m+p>2, take r=m+p-1. Then, (225) is ensured by the following inequalities:
(226)c2*B≤m(1-ε)q(a-ε)(1-ε(m+p-2)/2),(m+p-1)B2qε(m+p-2)/2≤m(1-ε)q(a-ε).
Take
(227)B=m(1-ε)q(a-ε)(1-ε(m+p-2)/2)c2*.
Then, when ε<(c2*2/m(m+p-1))2/(m+p-2), (225) holds. Recalling (219), we obtain
(228)-mac2*φm+p-1≤ψ(φ)≤-m(1-ε)q(a-ε)(1-ε(m+p-2)/2)c2*φm+p-1
when φc2*τ<ε. When p>1, we have
(229)c2*(1-ε)q(a-ε)(1-ε(m+p-2)/2)(p-1)≤limφ→0+ξ(φ)φp-1≤c2*a(p-1);
that is,
(230)((1-ε)q(a-ε)(1-ε(m+p-2)/2)(p-1)c2*)1/(1-p)≤limξ→+∞φ(ξ)ξ1/(p-1)≤(a(p-1)c2*)1/(1-p).
When p=1, we have
(231)-c2*(1-ε)q(a-ε)(1-ε(m+p-2)/2)≤limφ→0+ξ(φ)lnφ≤-c2*a;
that is,
(232)e-(a/c2*)ξ≤φ(ξ)≤e-((1-ε)q(a-ε)(1-ε(m+p-2)/2)/c2*)ξ.
By the arbitrariness of ε>0, (205)-(206) hold.
When1<m+p<2, take r=(m+p)/2. Then, (225) holds if
(233)m+p2B2≤m(1-ε)q(a-ε)(1-ε(2-m-p)/4),c2*Bε(2-m-p)/4≤m(1-ε)q(a-ε).
Take
(234)B=2mm+p(1-ε)q(a-ε)(1-ε(2-m-p)/4).
Then, when
(235)ε≤min{a2,(a2(1-a2)q)2/(2-m-p)×(m(m+p)2c2*-2)2/(2-m-p)},
(225) holds. Combining with (218), we obtain
(236)-2mam+pφ(m+p)/2≤ψ(φ)≤-2mm+p(1-ε)q(a-ε)(1-ε(2-m-p)/4)φ(m+p)/2
when φc2*τ<ε. When p>m, a simple calculation yields
(237)(p-m2m2m(1-ε)q(a-ε)(1-ε(2-m-p)/4)m+p)2/(m-p)≤limξ→+∞φ(ξ)ξ2/(p-m)≤(p-m2m2mam+p)2/(m-p).
When p=m,
(238)e-(a/m)ξ≤φ(ξ)≤e-((1-ε)q(a-ε)(1-ε(1-m)/2)/m)ξ.
By the arbitrariness of ε>0, (209)-(210) hold.
Whenm+p=2, take
(239)r=1,B=c2*2+4m(1-ε)q(a-ε)-c2*2.
Then, (225) holds. On the other hand, consider the following problem:
(240)12dv2dφ-c2*v≥maφ(1-φ)q,forφ≥0,v(0+)=0,v(φ)<0,forφ>0.
We can see that ψ(φ)≥v(φ). Let
(241)v=-c2*2+4ma-c2*2φ.
Then, v satisfies (240). Thus, we conclude that
(242)-c2*2+4ma-c2*2φ≤ψ(φ)≤-c2*2+4m(1-ε)q(a-ε)-c2*2φ.
When m<1, that is, p>1,
(243)((1-m)(c2*2+4m(1-ε)q(a-ε)-c2*)2m)1/(m-1)≤limξ→+∞φ(ξ)ξ1/(1-m)≤((1-m)(c2*2+4ma-c2*)2m)1/(m-1).
When m=1, that is, p=1,
(244)e-((c2*2+4a-c2*)/2)ξ≤φ(ξ)≤e-((c2*2+4(1-ε)q(a-ε)-c2*)/2)ξ.
By the arbitrariness of ε>0, (207)-(208) hold.


## 5. Discussion

Whenp=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].



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*Source: 578345-2013-04-18.xml*