# Exponential Stability of Traveling Waves for a Reaction Advection Diffusion Equation with Nonlinear-Nonlocal Functional Response

Authors: Rui Yan; Guirong Liu
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

The stability of a reaction advection diffusion equation with nonlinear-nonlocal functional response is concerned. By using the technical weighted energy method and the comparison principle, the exponential stability of all noncritical traveling waves of the equation can be obtained. Moreover, we get the rates of convergence. Our results improve the previous ones. At last, we apply the stability result to some real models, such as an epidemic model and a population dynamic model.


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## Body

## 1. Introduction

As an important nonlinear parabolic equation, reaction (advection) diffusion equations are usually used to describe the development of practical problems related to time and space distribution. The large number of problems involved in reaction-diffusion equations arises from a large number of mathematical models in chemistry, physics, ecology, biology, and infectious diseases, for instance, [1–4]. The traveling wave solutions of reaction-diffusion equations refer to the solutions of the shape, such as u(t,x)=u(x+ct), where the wave velocity c is a real constant. In practical applications, traveling wave solutions can well explain oscillations and finite velocity propagation phenomena in nature such as the change of the concentration of a reactant in a chemical reaction, the invasion of species in biology, the mutual transformation of two states of matter in condensed matter physics, the propagation of nerve impulses in neural networks, and the spread of infectious diseases. The stability of traveling wave solutions based on the existence of traveling wave equations is also very important and has great theoretical and practical value. In the phase transition process, a necessary condition that the state change can be observed is that the traveling wave solutions are stable. For example, in the sense of epidemiology, the traveling wave solutions describe the transition from a disease-free equilibrium to an endemic equilibrium, while the existence and nonexistence of nontrivial traveling wave solutions suggest whether or not the disease can spread. The results will help predict the developing trend of infectious diseases, identify the key factors for the spread of infectious diseases, and seek the best strategies to prevent and control the spread of infectious diseases.Hence, the study of traveling wave solutions has become a very interesting problem, in which the stability of traveling waves is significant and difficult, and this topic attracts enormous attention. For the stability of reaction-diffusion equations without time delay, there is a great deal of literature, for example, [5–9]. For the delayed reaction-diffusion equations, Schaaf [10] and the authors [11] studied the stability of the traveling waves. Then, by applying the upper and lower solutions method combined with the squeezing technique, Wang et al. [12] presented the exponential stability of traveling wavefronts for a reaction advection diffusion equation with spatiotemporal delays under the bistable assumptions. Mei et al. [13, 14] proved that the large waves were exponentially stable when an initial solution is close enough to the traveling wave solution under a weighted L2 norm. Later, Mei and coauthors [15, 16] obtained the globally exponential stability of the traveling waves by using the similar methods in [17] with some improvement. Recently, Mei et al. [18] gave the globally exponential stability of all noncritical wavefronts and the globally algebraic stability of critical wavefronts for a nonlocal time-delayed reaction-diffusion equations by using the weighted energy method and the Fourier transform. Immediately, Mei and Wang [19] obtained the similar results for a class of nonlocal time-delayed Fisher-KPP type reaction-diffusion equations in n-dimensional space by employing the similar methods, in which the convergence rates were more accurate than previous results.Motivated by these previous works, in this paper, we consider the stability of the traveling waves of the following reaction advection diffusion equation with nonlinear-nonlocal functional response(1)∂ut,x∂t=D∂2ut,x∂2x+B∂ut,x∂x+Fut,x,K*fut,x,t≥0,x∈R,with the initial data (2)us,x=u0s,x,s∈-r,0,x∈R,where B∈R,r≥0, D>0 is a diffusion coefficient, and the convolution is defined by (3)K*fut,x=∫0r∫-∞∞Ks,yfut-s,x-ydyds. Some well-known models can be obtained by choosing proper cases of F,f and K. For example, some of them can be generalized as below.(i) SetB=0,f(u)=u and K(s,y)=δ(s-r)δ(y); then (1) turns to the following local reaction-diffusion equation with discrete delay (4)utt,x=DΔut,x+Fut,x,ut-r,x.Schaaf [10] studied the stability of the traveling waves of (4) by a phase plane analysis method. Then, the authors in [11] showed the global asymptotic stability, Lyapunov stability, and uniqueness of traveling waves of (4) by using the elementary super- and subsolution comparison and squeezing methods.(ii) SetB=0 and K(s,y)=δ(s-r)J(y); then (1) turns to the following nonlocal reaction-diffusion equation with discrete time delay (5)utt,x=DΔut,x+Fut,x,∫-∞∞Jyfut-r,x-ydy.In [20], the existence of traveling wavefronts of (5) was investigated by using the sub- and supersolution method. The authors also proved the asymptotic stability and uniqueness of traveling wave fronts by applying the comparison principle and squeezing technique. Later, Wang [21] proved the existence of the traveling wave solutions of (5) by using the upper and lower solutions method and the Schauder fixed point theorem when the term f may not be monotone or quasimonotone. Recently, the authors [22] got the exponential stability of all noncritical traveling waves of (5) and the algebraic stability of critical traveling waves of (5) by using the weighted energy method coupled with the comparison principle.It follows from the above examples that these local or nonlocal equations with discrete time delay are studied intensively. In some cases, a discrete delay is a good approximation, but the distributed delay is necessary in others. On the other hand, reaction advection diffusion equations have been usually applied to describe the dispersion courses in the mobile media such as fluids; see, for example, [12, 23]. In fact, Wu [24] proved the exponential stability of traveling wavefronts with large wave velocity of (1) by using the comparison principle and the technical weighted energy method. By choosing a different weighted energy function from [24] and adopting different estimates, then we can get the stability of all noncritical traveling wavefronts of (1), not just the traveling waves with large wave velocity.Now we impose some assumptions on (1) as follows:(A1)
K(s,y) is a continuous nonnegative function with K(s,-y)=K(s,y) for s≥0,y∈R and ∫0r∫-∞∞Ks,ydyds=1, ∫0r∫-∞∞K(s,y)e-λ(y+cs)dyds<∞ for any c≥0 and λ≥0.(A2)
For anyt∈[0,r],x∈R,K(t,x)=K¯(t,Bt+x), where K¯ satisfies K¯(t,x)=K¯(t,-x).(A3)
LetK>0, F∈C2[0,K]×[0,f(K)],R, F(0,f(0))=F(K,f(K))=0,F(x,f(x))>0, for x∈(0,K),∂2F(x,y)>0,∂1F(0,0)<0,∂ijF(x,y)≤0(i,j=1,2) for any (x,y)∈[0,K]×[0,f(K)], where ∂1F(x,y)=∂F(x,y)/∂x and ∂2F(x,y)=∂F(x,y)/∂y.(A4)
f∈C2([0,K],R+),f(0)=0,f′(x)≥0,f′′(x)≤0 for any x∈[0,K].(A5)
f′0f′(K)∂2F(0,0)∂2F(K,0)-(∂1F(K,f(K)))2<0.(A6)
∂2FK,fKf′(K)+∂1F(K,f(K))<0.From(A3), it is easy to see that (1) has two constant equilibria u±, where (6)u-=0,u+=K.Throughout this paper, a traveling wave solution of (1) always refers to a pair (ϕ,c), where ϕ=ϕ(ξ)(ξ=x+ct) is a function on R satisfying the following ordinary differential equation: (7)c-Bϕ′ξ=Dϕ′′ξ+Fϕξ,∫0r∫-∞∞Ks,yfϕξ-y-csdyds,ϕ±∞=u±.We call c the traveling wave speed.The rest of the paper is organized as follows. In Section2, we give some preliminaries and present our main result on the exponential stability of traveling waves of (1). In Section 3, we will give the proof the main result by using the weighted energy method. In Section 4, we apply our results to some models, such as an epidemic model and a population dynamic model.


## 2. Preliminaries and Main Results

In this paper, we suppose thatC>0 represents a generic constant and Ci>0 denotes a particular constant. Set I as an interval; typically I=R. Take (8)LpI=fx:∫Ifpxdx<∞,x∈I,p≥1,LωpI=fx:∫Iωxfpxdx<∞,x∈I,p≥1,ωx≥0,HkI=fx∈L2I:difdxi∈L2I,i=1,…,k,x∈I with the norm defined by (9)fLωp=∫Iωxfxpdx1/p,p≥1,fHkI=∑i=0k∫Iωxdidxifx2dx1/2. Moreover, set T>0 as a constant and B as a Banach space; we denote by C([0,T];B) the space of the B-valued continuous functions on [0,T]. The corresponding spaces of the B-valued functions on [0,∞) are defined similarly.Now we show the following existence result of traveling wave solutions of (1).Theorem 1 ([24], existence of traveling waves).
Assume that(A1)–(A4) hold; there exist numbers c∗>0 and λ∗(c∗)>0 satisfying (10)Pλ∗,c∗+B=0,∂∂λPλ,c∗+Bλ∗=0,where (11)Pλ,c=Dλ2-c-Bλ+∂1F0,0+∂2F0,0f′0∫0r∫-∞∞Ks,ye-λy+csdyds.For every c>c∗+B, (1) has a monotone traveling wave solution ϕ(x+ct) such that ϕ(-∞)=0 and ϕ(+∞)=K.From the assumptions we can introduce that∂1F(0,0)+∂2F0,0f′(0)>0. Then by a simple calculation similar to that of [21] we can get the following conclusions. When c>c∗+B, P(λ,c)=0 has two positive real roots λ1(c) and λ2(c) with 0<λ1(c)<λ2(c) which also satisfies (12)Pλ,c<0forλ1<λ<λ2.When c=c∗+B, it holds that (13)Pλ∗,c∗+B=0forλ1=λ∗=λ2.Then, forc>c∗+B, we define a weight function related to such a number λ>0, (14)ωξ=e-λξ-ξ0,ξ≤ξ0,1,ξ>ξ0 with a sufficiently large number ξ0≫1, where λ∈(λ1(c),λ∗]. Obviously ω(ξ)≥1 for all ξ∈R and ω(ξ)→+∞ as ξ→-∞.Then we give the following globally exponential stability of the traveling wavefronts of (1).Theorem 2 (stability of traveling waves).
Suppose that(A1)–(A6) hold. For the traveling wave ϕ(ξ) of (1) and ϕ(±∞)=u± with speed c>c∗+B, if the initial data holds (15)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×Rand the initial perturbation is u0(s,x)-ϕ(x+cs)∈C[-r,0];Lω1(R)∩H1(R); then the solution of (1) and (2) satisfies (16)0=u-≤ut,x≤u+=K,fort,x∈0,+∞×R,ut,x-ϕx+ct∈C0,+∞;Lω1R∩H1R. If c>c∗+B, the solution u(t,x) converges to the traveling wave ϕ(x+ct) exponentially (17)supx∈R⁡ut,x-ϕx+ct≤Ce-μt,t≥0,with a positive number μ, where μ=μ1/3 and μ1 is determined by (18)c-Bλ-Dλ2-∂1F0,0-μ1-∂2F0,0f′0eμ1rMr,λ>0,λ∈λ1c,λ∗,Mr,λ=∫0r∫-∞∞Kτ,ye-λy+cτdydτ,-∂1FK,fK-μ1-∂2FK,fKf′Keμ1r>0.Remark 3.
The authors in [24] investigated the exponential stability of traveling waves of (1) when (19)c>max⁡c∗+B,2Dc∗2+LL0L2+2L1+B, where (20)L0=maxu∈0,K⁡f′u,Li=maxu,v∈0,K×0,fK⁡∂iFu,v,i=1,2,L=∫0r∫-∞∞Kτ,ye-c∗/Dy+cτdydτ.In addition, [24] needs the following assumption: (21)2∂1FK,fK+∂2FK,fKf′K<-L∂2FK,fKf′K.But condition (A5) is not required in [24]. In Section 4, we can demonstrate the advantage of our result by the same example.


## 3. Proof of the Stability

For the stability result, we need the following boundedness and the comparison principle for (1). The proofs are similar to those of [16, 24], so we omit them here.Lemma 4 (boundedness).
Assume(A1)–(A4) hold and the initial data satisfy (22)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×R; thus the solution u(t,x) of the Cauchy problem (1) and (2) exists uniquely and satisfies (23)0=u-≤ut,x≤u+=K,fort,x∈0,+∞×R.Lemma 5 (comparison principle).
Setu¯(t,x) and u_(t,x) as the solutions of (1) and (2) with the initial data u¯0(s,x) and u_0(s,x), respectively. When (24)0=u-≤u_0s,x≤u¯0s,x≤u+=K,fors,x∈-r,0×R, thus (25)0=u-≤u_t,x≤u¯t,x≤u+=K,fort,x∈0,+∞×R.Ifu0(s,x) satisfies (26)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×R, letting (27)U0+s,x=max⁡u0s,x,ϕx+cs,fors,x∈-r,0×R,U0-s,x=min⁡u0s,x,ϕx+cs,fors,x∈-r,0×R,then (28)u-≤U0-s,x≤u0s,x≤U0+s,x≤u+,fors,x∈-r,0×R,u-≤U0-s,x≤ϕx+cs≤U0+s,x≤u+,fors,x∈-r,0×R.Set U+(t,x) and U-(t,x) as the corresponding solutions of (1) and (2) with initial data U0+(s,x) and U0-(s,x) determined in (27); that is, (29)Ut±t,x=DΔU±t,x+FU±t,x,∫0r∫-∞∞Ks,yfU±t-s,x-ydyds,t≥0,x∈R,U±s,x=U0±s,x,s,x∈-r,0×R. Thus by Lemma 5 we get (30)u-≤U-t,x≤ut,x≤U+t,x≤u+,fort,x∈0,+∞×R,u-≤U-t,x≤ϕx+ct≤U+t,x≤u+,fort,x∈0,+∞×R.Now we need the following three steps to get the stability result.Step 1.
The convergence ofU+(t,x) to ϕ(x+ct) is as follows.
For everyc>c∗+B, take ξ=x+ct and (31)vt,ξ=U+t,x-ϕx+ct,v0s,ξ=U0+s,x-ϕx+cs.From (28) and (30) we get (32)vt,ξ≥0,v0s,ξ≥0.Then it can be checked that v(t,ξ) defined in (31) satisfies (33)vtt,ξ+c-Bvξt,ξ-Dvξξt,ξ-∂1Fξvt,ξ=∂2FξΦ~ξ+Qwith the initial data (34)vs,ξ=v0s,ξ,s,ξ∈-r,0×R,where (35)Φ^ξ=∫0r∫-∞∞Ks,yfϕξ-y-csdyds,Φ~ξ=∫0r∫-∞∞Ks,yf′ϕξ-y-csvt-s,ξ-y-csdyds,Φ¯ξ=∫0r∫-∞∞Ks,yfvt-s,ξ-y-cs+ϕξ-y-csdyds,Fξ=Fϕξ,Φ^ξ,(36)Q=Fvt,ξ+ϕξ,Φ¯ξ-Fξ-∂1Fξvt,ξ-∂2FξΦ~ξ.Lemma 6.
Forω1=e-λ(ξ-ξ0)(λ∈(λ1(c),λ∗]), by choosing μ1>0 such that (37)c-Bλ-Dλ2-∂1F0,0-μ1-∂2F0,0f′0eμ1rMr,λ=C1>0,where (38)Mr,λ=∫0r∫-∞∞Kτ,ye-λy+cτdydτ,then, for c>c∗+B, the following holds: (39)vtLω11R+C1∫0t∫-∞∞e-μ1t-sω1ξvs,ξdξds≤Ce-μ1t.Proof.
Assume the solution of (33)-(34) is mollifying enough. Multiplying (33) by ω1(ξ)eμ1t, where μ1 is defined in (37), we get (40)eμ1tω1vt+eμ1tc-Bω1v-Dω1vξ+Dω1′vξ-μ1eμ1tω1v+eμ1t-c-Bω1′-Dω1′′-∂1Fξω1v=eμ1tω1∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-csvs-τ,ξ-y-cτdydτ+Q.Integrating both sides of (40) over R×[0,t] with respect to ξ and t, thus (41)eμ1t∫-∞∞ω1ξvt,ξdξ+∫0t∫-∞∞eμ1s-c-Bω1′-Dω1′′-∂1Fξω1-μ1ω1vdξds=v00Lω11R+∫0t∫-∞∞eμ1sω1ξQs,ξdξds+∫0t∫-∞∞eμ1sω1ξ∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-csvs-τ,ξ-y-cτdydτdξds.For Q, applying Taylor’s formula to (36) and noting (A3)-(A4); then (42)Q≤12∂11Fϕ¯,ϕ~v2+∂12Fϕ¯,ϕ~Φ¯-Φ^v+12∂22Fϕ¯,ϕ~Φ¯-Φ^2≤0,where ϕ¯ is some function between ϕ and ϕ+v and ϕ~ is some function between Φ¯ and Φ^.
For the third term on the right-hand side of (41) by Fubini’s theorem and making the change of variables ξ-y-cτ→ξ,s-τ→s,y→y,τ→τ, it follows that (43)∫0t∫-∞∞eμ1sω1ξ∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdydτdξds=∫0r∫-∞∞∫0t∫-∞∞eμ1sω1ξ∂2FξKτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdξdsdydτ=∫0r∫-∞∞∫-τt-τ∫-∞∞eμ1s+τω1ξ+y+cτ∂2Fξ+y+cτKτ,yf′ϕξvs,ξdξdsdydτ≤eμ1r∫0t∫-∞∞∫0r∫-∞∞eμ1sω1ξ+y+cτ∂2Fξ+y+cτKτ,yf′ϕξvs,ξdydτdξds+eμ1r∫-r0∫-∞∞∫0r∫-∞∞eμ1sω1ξ+y+cτ∂2Fξ+y+cτKτ,yf′ϕξv0s,ξdydτdξds.Substituting (42) and (43) into (41), it follows that (44)eμ1t∫-∞∞ω1ξvt,ξdξ+∫0t∫-∞∞eμ1sω1ξvs,ξAc,μ1,ω1ξdξds≤v00Lω11R+eμ1r∫-r0∫-∞∞∫0r∫-∞∞eμ1sω1ξ+y+cτ∂2Fξ+y+cτKτ,yf′ϕξv0s,ξdydτdξds,where (45)Ac,μ1,ω1ξ=-c-Bω1′ξω1ξ-Dω1′′ξω1ξ-∂1Fξ-μ1-eμ1r∫0r∫-∞∞ω1ξ+y+cτω1ξ∂2Fξ+y+cτKτ,yf′ϕξdydτ.Since ω1′(ξ)=-λe-λ(ξ-ξ0),ω1′′(ξ)=λ2e-λ(ξ-ξ0),ω1(ξ+y+cτ)/ω1(ξ)=e-λ(y+cτ) and noting (A5), the following holds: (46)Ac,μ1,ω1ξ≥c-Bλ-Dλ2-∂1F0,0-μ1-eμ1r∂2F0,0f′0∫0r∫-∞∞Kτ,ye-λy+cτdydτ. Using the fact that, for c>c∗+B,λ1<λ≤λ∗, (47)c-Bλ-Dλ2-∂1F0,0-∂2F0,0f′0∫0r∫-∞∞Kτ,ye-λy+cτdydτ>0, then we can choose a small μ1>0 such that (48)Ac,μ1,ω1ξ≥c-Bλ-Dλ2-∂1F0,0-μ1-eμ1r∂2F0,0f′0∫0r∫-∞∞Kτ,ye-λy+cτdydτ=C1>0.For the second term on the right-hand side of (44), we have (49)eμ1r∫-r0∫-∞∞∫0r∫-∞∞eμ1sω1ξ+y+cτω1ξ∂2Fξ+y+cτKτ,yf′ϕξω1ξv0s,ξdydτdξds≤eμ1r∂2F0,0f′0∫-r0∫-∞∞eμ1sω1ξv0s,ξ∫0r∫-∞∞Kτ,ye-λy+cτdydτdξds≤C∫-r0v0sLω11Rds.Substituting (48)-(49) into (44) we can get (50)eμ1t∫-∞∞ω1ξvt,ξdξ+C1∫0t∫-∞∞eμ1sω1ξvs,ξdξds≤C;that is, (51)vtLω11R+C1∫0t∫-∞∞e-μ1t-sω1ξvs,ξdξds≤Ce-μ1t,forc>c∗+B. This completes the proof.Lemma 7.
For everyc>c∗+B,t≥0, there holds that (52)vtL2R2+∫0tvsH1R2ds≤C.Proof.
In view ofω1(ξ)=e-λ(ξ-ξ0)≥1 for ξ∈(-∞,ξ0], it can follow from (50) that, for any c>c∗+B, (53)∫-∞ξ0vt,ξdξ+C1∫0t∫-∞ξ0vs,ξdξds≤C∀t≥0, and letting t→∞, we obtain (54)∫0∞∫-∞ξ0vs,ξdξds≤C. Since 0≤v(t,ξ)≤u+=K, for (t,ξ)∈R+×R, we have (55)∫0∞∫-∞ξ0v2s,ξdξds≤u+∫0∞∫-∞ξ0vs,ξdξds≤C.Multiplying (33) by v(t,ξ) and integrating it over R×[0,t] with respect to ξ and t, it yields that (56)vtL2R2+2D∫0tvξs,ξL2R2ds-2∫0t∫-∞∞∂1Fξv2s,ξdξds=v00L2R2+2∫0t∫-∞∞Q·vs,ξdξds+2∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdydτvs,ξdξds.By employing the Cauchy inequality 2xy≤ηx2+1/η·y2 for η>0, we get (57)2∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdydτvs,ξdξds≤η∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτv2s,ξdydτdξds+1η∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτv2s-τ,ξ-y-cτdydτdξds.For the second term on the right-hand side of (57), by making the change of variables ξ-y-cτ→ξ,s-τ→s,y→y,τ→τ and using (55), we have (58)1η∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτv2s-τ,ξ-y-cτdydτdξds=1η∫0r∫-∞∞∫0t∫-∞∞∂2FξKτ,yf′ϕξ-y-cτv2s-τ,ξ-y-cτdξdsdydτ=1η∫0r∫-∞∞∫-τt-τ∫-∞∞∂2Fξ+y+cτKτ,yf′ϕξv2s,ξdξdsdydτ≤1η∫-r0∫-∞∞∫0r∫-∞∞∂2Fξ+y+cτKτ,yf′ϕξv2s,ξdydτdξds+1η∫0t∫-∞ξ0∫0r∫-∞∞∂2Fξ+y+cτKτ,yf′ϕξv2s,ξdydτdξds+1η∫0t∫ξ0∞∫0r∫-∞∞∂2Fξ+y+cτKτ,yf′ϕξv2s,ξdydτdξds≤1η∂2F0,0f′0∫-r0∫-∞∞v02s,ξdξds+1η∂2F0,0f′ϕξ0∫0t∫ξ0∞v2s,ξdξds+C=1η∂2F0,0f′ϕξ0∫0t∫ξ0∞v2s,ξdξds+C.For the first term on the right-hand side of (57), by using (55), we have (59)η∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτv2s,ξdydτdξds≤ηf′0∂2Fϕξ0,0∫0t∫ξ0∞v2s,ξdξds+C.Substituting (58), (59) into (57), we obtain (60)2∫0t∫-∞∞∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdydτvs,ξdξds≤ηf′0∂2Fϕξ0,0+1η∂2F0,0f′ϕξ0∫0t∫ξ0∞v2s,ξdξds+C.Noticing that v(t,ξ)≥0,Q≤0 and using (60) to (56), it yields that (61)vtL2R2+2D∫0tvξs,ξL2R2ds-2∫0t∫-∞ξ0∂1Fξv2s,ξdξds-∫0t∫ξ0∞2∂1Fξ+ηf′0∂2Fϕξ0,0+1η∂2F0,0f′ϕξ0v2s,ξdξds≤C.Because of (A5) and (A6), we can select η>0 as follows:(62)0<-∂1FK,fK-∂12FK,fK-f′0∂2FK,0∂2F0,0f′Kf′0∂2FK,0<η<-∂1FK,fK+∂12FK,fK-f′0∂2FK,0∂2F0,0f′Kf′0∂2FK,0. Then we have (63)2∂1FK,fK+ηf′0∂2FK,0+1η∂2F0,0f′K<0. Choose ξ0 large enough such that if ϕξ0-u+≪1, we can obtain (64)C2≔-2∂1Fξ0+ηf′0∂2Fϕξ0,0+1η∂2F0,0f′ϕξ0>0. Because of ∂ijF(x,y)≤0, for ξ∈[ξ0,∞), we have (65)2∂1Fϕξ,Φ^ξ+ηf′0∂2Fϕξ0,0+1η∂2F0,0f′ϕξ0≤2∂1Fϕξ0,Φ^ξ0+ηf′0∂2Fϕξ0,0+1η∂2F0,0f′ϕξ0=-C2<0.Thus substituting (65) into (61) and setting C3=min⁡{C2,-2∂1F(0,0)} we can obtain (66)vtL2R2+2D∫0tvξs,ξL2R2ds+C3∫0tvsL2R2ds≤C. This completes the proof.Next, we differentiate (33) with respect to ξ, multiply it by vξ(t,ξ), and integrate it over [0,t]×R with respect to t and ξ. Combining with Lemma 7, we can get the following energy estimate. Here we omit the details of the proof.Lemma 8.
Ifc>c∗+B and t≥0, the following holds: (67)vξtL2R2+∫0tvξsH1R2ds≤C.Lemma 9.
Ifc>c∗+B and t≥0, the following holds:(68)vtL∞-∞,ξ0≤Ce-μ1t/3.Proof.
TakeI=(-∞,ξ0]. Thus (69)vtL2I2=∫-∞ξ0vt,ξ2dξ≤vtL∞IvtL1I,(70)v2t,ξ=∫-∞ξ∂ξv2dξ=2∫-∞ξvt,ξvξt,ξdξ. By the Hölder inequality, we can get (71)vtL∞I2≤2∫-∞ξ0v2t,ξdξ1/2∫-∞ξ0vξ2t,ξdξ1/2=2vtL2IvξtL2I.It follows from (69) and (71) that (72)vtL∞I2≤2vtL∞I1/2vtL1I1/2vξtL2I; that is, (73)vtL∞I≤43vtL1I1/3vξtL2I2/3.From (67) we have (74)vξtL2R≤C.Noting that ω1(ξ)=e-λ(ξ-ξ0)≥1 if ξ∈I and combining with (39), we can get (75)vtL1I≤vtLω11I≤Ce-μ1t,t≥0.Then it is obvious from (73)–(75) that (76)vtL∞I≤Ce-μ1t/3,t≥0,vt,ξ0≤Ce-μ1t/3,t≥0. This completes the proof.Next, we need to prove the convergence of the traveling wave solutions in another interval[ξ0,+∞).Lemma 10.
Ifc>c∗+B and t≥0, we have (77)vtL∞ξ0,∞≤Ce-μ1t/3.Proof.
Similar to front, we multiply (33) by eμ1t and integrate it over [0,t]×R with respect to (t,ξ). Since Q≤0, we can get (78)eμ1tvtL1R-∫0t∫-∞∞eμ1s∂1Fξ+μ1vs,ξdξds-∫0t∫-∞∞eμ1s∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdydτdξds≤v00L1R.Similar to (43), we have (79)∫0t∫-∞∞eμ1s∂2Fξ∫0r∫-∞∞Kτ,yf′ϕξ-y-cτvs-τ,ξ-y-cτdydτdξds≤eμ1r∫0t∫-∞∞∫0r∫-∞∞eμ1s∂2Fξ+y+cτKτ,yf′ϕξvs,ξdydτdξds+eμ1r∫-r0∫-∞∞∫0r∫-∞∞eμ1s∂2Fξ+y+cτKτ,yf′ϕξv0s,ξdydτdξds.Substituting (79) into (78), it follows that (80)eμ1tvtL1R-∫0t∫-∞∞eμ1s∂1Fξ+μ1vs,ξdξds-eμ1r∫0t∫-∞∞∫0r∫-∞∞eμ1s∂2Fξ+y+cτKτ,yf′ϕξvs,ξdydτdξds-eμ1r∫-r0∫-∞∞∫0r∫-∞∞eμ1s∂2Fξ+y+cτKτ,yf′ϕξv0s,ξdydτdξds≤v00L1R.Since R=(-∞,ξ0]∪[ξ0,∞), we obtain (81)eμ1t∫ξ0∞vt,ξdξ-∫0t∫ξ0∞eμ1s∂1Fξ+μ1vs,ξdξds-eμ1r∫0t∫ξ0∞eμ1sf′ϕξvs,ξ∫0r∫-∞∞∂2Fξ+y+cτKτ,ydydτdξds≤v00L1R-It,where (82)It=eμ1t∫-∞ξ0vt,ξdξ-∫0t∫-∞ξ0eμ1s∂1Fξ+μ1vs,ξdξds-eμ1r∫0t∫-∞ξ0eμ1sf′ϕξvs,ξ∫0r∫-∞∞∂2Fξ+y+cτKτ,ydydτdξds+eμ1r∫-r0∫-∞∞∫0r∫-∞∞eμ1s∂2Fξ+y+cτKτ,yf′ϕξv0s,ξdydτdξds. Because ω1(ξ)≥1,ξ∈(-∞,ξ0] and combining with (39), then we have (83)eμ1t∫-∞ξ0vt,ξdξ≤C,∫0t∫-∞ξ0eμ1svs,ξdξds≤C. Also in view of (A1),(A3),(A4) and (84)∫-r0∫-∞∞eμ1sv0s,ξdξds≤∫-r0∫-∞∞v0s,ξdξds, then it yields that (85)It≤C,fort≥0.Substituting (85) into (81), we have (86)eμ1t∫ξ0∞vt,ξdξ-∫0t∫ξ0∞∂1Fξ+μ1+f′ϕξeμ1r∫0r∫-∞∞∂2Fξ+y+cτKτ,ydydτeμ1svs,ξdξds≤C.In view of the assumption ∂2FK,fKf′(K)+∂1F(K,f(K))<0, thus there is a constant μ1>0 small enough such that (87)-∂1FK,fK-μ1-∂2FK,fKf′Keμ1r>0. Since limξ→+∞ϕ(ξ)=K,limξ→+∞Φ^(ξ)=f(K), then when ξ0≫1, we can get (88)-∂1Fξ-μ1-f′ϕξeμ1r∫0r∫-∞∞∂2Fξ+y+cτKτ,ydydτ>0,ξ∈ξ0,∞.Putting (88) to (86), it yields that (89)∫ξ0∞vt,ξdξ≤Ce-μ1t. Similar to (68), it follows that (90)vtL∞ξ0,∞≤43vtL1ξ0,∞1/3vξtL2ξ0,∞2/3≤Ce-μ1t/3. This completes the proof.From Lemmas9 and 10, we can get the following result directly.Lemma 11.
Ifc>c∗+B and t≥0, it yields that (91)supx∈R⁡U+t,x-ϕx+ct≤Ce-μt, where μ=μ1/3.Step 2.
The convergence ofU-(t,x) to ϕ(x+ct) is as follows.
For everyc>c∗+B, take ξ=x+ct and (92)vt,ξ=ϕx+ct-U-t,x,v0s,ξ=ϕx+cs-U0-s,x. Similar to process 1, we can get the following convergence of U-(t,x) to ϕ(x+ct).Lemma 12.
Ifc>c∗+B and t≥0, the following holds: (93)supx∈R⁡U-t,x-ϕx+ct≤Ce-μt.Step 3.
The convergence ofu(t,x) to ϕ(x+ct) is as follows.Lemma 13.
Ifc>c∗+B and t≥0, the following holds: (94)supx∈R⁡ut,x-ϕx+ct≤Ce-μt.Proof.
Because ofU0-(s,x)≤u0(s,x)≤U0+(s,x), by Lemma 5, we can get U-(t,x)≤u(t,x)≤U+(t,x),(t,x)∈R+×R. Combining with Lemmas 11 and 12 and applying the squeeze theorem, it then follows that (95)supx∈R⁡ut,x-ϕx+ct≤Ce-μt,fort≥0,c>c∗+B. This completes the proof.


## 4. Applications

In this section, we will apply the content of Theorem2 to some realistic examples.


### 4.1. A Vector-Disease Model

In [3, 25], the authors considered the following equation: (96)utt,x=DΔut,x-aut,x+b1-ut,x∫0r∫-∞∞Ks,yut-s,x-ydyds,where b>a>0. This equation presented a host-vector model for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Here u(t,x) is the density of infectious individual at time t and at position x, a denotes the recovery rate of the infected individual, and b denotes the host-vector contact rate. D denotes the diffusion constant.Clearly, (96) has two constant equilibria u0≡0 and u+≡1-a/b>0. The corresponding characteristic equation at u0≡0 is (97)Pλ,c=Dλ2-cλ-a+b∫0r∫-∞∞Ks,ye-λy+csdyds. Set (98)c∗=inf⁡c>0∣Pλ,c=0has  a  positive  real  root.There exists a corresponding number λ∗>0 such that P(λ∗,c∗)=0. If c>c∗, P(λ,c) has two real roots 0<λ1(c)<λ2(c).Now setF(x,y)=-ax+b(1-x)y,B=0 and f(x)=x in (1). It is easy to see that ∂1F(x,y)=-a-by<0, ∂2F(x,y)=b-bx>0, and ∂ijF(x,y)≤0 for (x,y)∈[0,u+]×[0,u+]. Also we have (99)f′0f′K∂2F0,0∂2FK,0-∂12FK,fK<0,∂2FK,fKf′K+∂1FK,fK<0,where K=1-a/b. Thus we get the following result.Corollary 14.
Suppose that(A1)-(A2) hold. For the given traveling wave ϕ(ξ) of (96) and ϕ(±∞)=u± with speed c>c∗, if the initial data satisfies (100)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×Rand the initial perturbation is u0(s,x)-ϕ(x+cs)∈C[-r,0];Lω1(R)∩H1(R), then the solution of (96) satisfies (101)0=u-≤ut,x≤u+=K,fort,x∈0,+∞×R,ut,x-ϕx+ct∈C0,+∞;Lω1R∩H1R. When c>c∗, the solution u(t,x) converges to the traveling wave ϕ(x+ct) exponentially (102)supx∈R⁡ut,x-ϕx+ct≤Ce-μt,t≥0,with a positive number μ=μ1/3, where μ1 is determined by (103)cλ-Dλ2+a-μ1-beμ1r∫0r∫-∞∞Ks,ye-λy+csdyds>0,λ∈λ1c,λ∗,b-μ1-aeμ1r>0.Remark 15.
In [24], under the additional condition b>1+1/2La, the exponential stability of traveling waves of (96) is obtained for (104)c>max⁡c∗,2Dc∗2+Lb-2a,where L=∫0r∫-∞∞e-c∗/Dy+csKs,ydyds.Remark 16.
Corollary14 shows that if b>a>0, the density of infectious individual tends to an endemic level u+=1-a/b when time t goes on. Further, when a↗b, the density of infectious individual tends to 0. That is, when the recovery rate of the infected individual is close to the host-vector contact rate, this disease will eventually die out. This shows that an effective way to control the epidemic is to reduce the host-vector contact rate. Therefore, the results will help predict the developing trend of infectious diseases, identify the key factors for the spread of infectious diseases, and seek the best strategies to prevent and control the spread of infectious diseases.


### 4.2. A Population Dynamic Model

In [17], the authors considered a single species in the spatial transport field and obtained the following general reaction advection diffusion equation with the nonlocal delayed effect if the mature mortality and diffusion rates are independent of age, (105)∂u∂t=∂∂xDm∂u∂x+Bu-dmu+ε∫-∞∞Φαx+Br-ybut-r,ydy,where (106)Φαx=14παe-x2/4α.Here u(t,x) is the density of the population of the species at time t and location x, Dm is the diffusion rate and dm is the mortality, B is the transport velocity of the field, r≥0 is the maturation time for the species, ε represents the impact of premature mortality, and α demonstrates the effect of immature population dispersal rate on mature population growth rate. Liang and Wu [17] obtained the existence of the traveling wavefronts by using the upper and lower solutions when b(u) took three different birth functions which have been widely studied in the well-known Nicholson’s blowflies equation. Finally, they gave the numerical stability of travelling waves by choosing some special data. Later, Wu et al. [24] showed the exponential stability of traveling waves with faster wave speed for (105). Now we will establish the exponential stability of all noncritical traveling waves for (105). First, we need to give the following hypotheses: (H1)
LetK>0,b(0)=0,εb(K)=dmK,b(x)∈C2[0,K],b′(x)≥0,b′′(x)≤0 for x∈[0,K], εb(x)>dmx for x∈(0,K).(H2)
ε2b′(0)b′(K)<dm2,εb′(K)<dm.Clearly,u-=0 and u+=K are two constant equilibria of (105). And the characteristic equation of (105) at u-=0 is (107)Pλ,c=Dmλ2-c-Bλ-dm+εb′0e-λcre4B2+4Bλ+λ2-4Br2/16α. There are c∗>0 and λ∗>0 which satisfy (108)Pλ∗,c∗+B=0,∂∂λPλ,c∗+Bλ∗=0. If c>c∗+B, P(λ,c) has two positive real roots 0<λ1(c)<λ2(c).TakingF(x,y)=-dmx+εy,f(x)=b(x),K(s,y)=Φα(y+Bs)δ(s-r) in (1), then (1) turns to (105). Now we show the stability of traveling wave solutions of (105).Corollary 17.
Assume(H1)-(H2) hold. For the given traveling wave ϕ(ξ) of (105) and ϕ(±∞)=u± with speed c>c∗+B, if the initial data holds (109)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×Rand the initial perturbation is u0(s,x)-ϕ(x+cs)∈C[-r,0];Lω1(R)∩H1(R), then the solution of (105) satisfies (110)0=u-≤ut,x≤u+=K,fort,x∈0,+∞×R,ut,x-ϕx+ct∈C0,+∞;Lω1R∩H1R. When c>c∗+B, the solution u(t,x) converges to the traveling wave ϕ(x+ct) exponentially (111)supx∈R⁡ut,x-ϕx+ct≤Ce-μt,t≥0,with a positive constant μ=μ1/3, where μ1 is determined by (112)c-Bλ-Dmλ2+dm-μ1-εb′0e-λcre4B2+4Bλ+λ2-4Br2/16α>0,dm-μ1-εb′Keμ1r>0.Remark 18.
The model can be used to study the behaviour of the mature population. For example, if we take birth functionb(u)=pue-auq, where p,a,q are positive constants and let 1<εp/dm<e1/q, it is easy to verify that, for c>c∗+B, all noncritical traveling waves connecting 0 and 1/aln⁡εp/dm1/q are exponential stable. That is, the density of the population tends to a constant 1/aln⁡εp/dm1/q as time becomes large. The results show that the density of the population will eventually reach a stable state.


## 4.1. A Vector-Disease Model

In [3, 25], the authors considered the following equation: (96)utt,x=DΔut,x-aut,x+b1-ut,x∫0r∫-∞∞Ks,yut-s,x-ydyds,where b>a>0. This equation presented a host-vector model for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Here u(t,x) is the density of infectious individual at time t and at position x, a denotes the recovery rate of the infected individual, and b denotes the host-vector contact rate. D denotes the diffusion constant.Clearly, (96) has two constant equilibria u0≡0 and u+≡1-a/b>0. The corresponding characteristic equation at u0≡0 is (97)Pλ,c=Dλ2-cλ-a+b∫0r∫-∞∞Ks,ye-λy+csdyds. Set (98)c∗=inf⁡c>0∣Pλ,c=0has  a  positive  real  root.There exists a corresponding number λ∗>0 such that P(λ∗,c∗)=0. If c>c∗, P(λ,c) has two real roots 0<λ1(c)<λ2(c).Now setF(x,y)=-ax+b(1-x)y,B=0 and f(x)=x in (1). It is easy to see that ∂1F(x,y)=-a-by<0, ∂2F(x,y)=b-bx>0, and ∂ijF(x,y)≤0 for (x,y)∈[0,u+]×[0,u+]. Also we have (99)f′0f′K∂2F0,0∂2FK,0-∂12FK,fK<0,∂2FK,fKf′K+∂1FK,fK<0,where K=1-a/b. Thus we get the following result.Corollary 14.
Suppose that(A1)-(A2) hold. For the given traveling wave ϕ(ξ) of (96) and ϕ(±∞)=u± with speed c>c∗, if the initial data satisfies (100)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×Rand the initial perturbation is u0(s,x)-ϕ(x+cs)∈C[-r,0];Lω1(R)∩H1(R), then the solution of (96) satisfies (101)0=u-≤ut,x≤u+=K,fort,x∈0,+∞×R,ut,x-ϕx+ct∈C0,+∞;Lω1R∩H1R. When c>c∗, the solution u(t,x) converges to the traveling wave ϕ(x+ct) exponentially (102)supx∈R⁡ut,x-ϕx+ct≤Ce-μt,t≥0,with a positive number μ=μ1/3, where μ1 is determined by (103)cλ-Dλ2+a-μ1-beμ1r∫0r∫-∞∞Ks,ye-λy+csdyds>0,λ∈λ1c,λ∗,b-μ1-aeμ1r>0.Remark 15.
In [24], under the additional condition b>1+1/2La, the exponential stability of traveling waves of (96) is obtained for (104)c>max⁡c∗,2Dc∗2+Lb-2a,where L=∫0r∫-∞∞e-c∗/Dy+csKs,ydyds.Remark 16.
Corollary14 shows that if b>a>0, the density of infectious individual tends to an endemic level u+=1-a/b when time t goes on. Further, when a↗b, the density of infectious individual tends to 0. That is, when the recovery rate of the infected individual is close to the host-vector contact rate, this disease will eventually die out. This shows that an effective way to control the epidemic is to reduce the host-vector contact rate. Therefore, the results will help predict the developing trend of infectious diseases, identify the key factors for the spread of infectious diseases, and seek the best strategies to prevent and control the spread of infectious diseases.


## 4.2. A Population Dynamic Model

In [17], the authors considered a single species in the spatial transport field and obtained the following general reaction advection diffusion equation with the nonlocal delayed effect if the mature mortality and diffusion rates are independent of age, (105)∂u∂t=∂∂xDm∂u∂x+Bu-dmu+ε∫-∞∞Φαx+Br-ybut-r,ydy,where (106)Φαx=14παe-x2/4α.Here u(t,x) is the density of the population of the species at time t and location x, Dm is the diffusion rate and dm is the mortality, B is the transport velocity of the field, r≥0 is the maturation time for the species, ε represents the impact of premature mortality, and α demonstrates the effect of immature population dispersal rate on mature population growth rate. Liang and Wu [17] obtained the existence of the traveling wavefronts by using the upper and lower solutions when b(u) took three different birth functions which have been widely studied in the well-known Nicholson’s blowflies equation. Finally, they gave the numerical stability of travelling waves by choosing some special data. Later, Wu et al. [24] showed the exponential stability of traveling waves with faster wave speed for (105). Now we will establish the exponential stability of all noncritical traveling waves for (105). First, we need to give the following hypotheses: (H1)
LetK>0,b(0)=0,εb(K)=dmK,b(x)∈C2[0,K],b′(x)≥0,b′′(x)≤0 for x∈[0,K], εb(x)>dmx for x∈(0,K).(H2)
ε2b′(0)b′(K)<dm2,εb′(K)<dm.Clearly,u-=0 and u+=K are two constant equilibria of (105). And the characteristic equation of (105) at u-=0 is (107)Pλ,c=Dmλ2-c-Bλ-dm+εb′0e-λcre4B2+4Bλ+λ2-4Br2/16α. There are c∗>0 and λ∗>0 which satisfy (108)Pλ∗,c∗+B=0,∂∂λPλ,c∗+Bλ∗=0. If c>c∗+B, P(λ,c) has two positive real roots 0<λ1(c)<λ2(c).TakingF(x,y)=-dmx+εy,f(x)=b(x),K(s,y)=Φα(y+Bs)δ(s-r) in (1), then (1) turns to (105). Now we show the stability of traveling wave solutions of (105).Corollary 17.
Assume(H1)-(H2) hold. For the given traveling wave ϕ(ξ) of (105) and ϕ(±∞)=u± with speed c>c∗+B, if the initial data holds (109)0=u-≤u0s,x≤u+=K,fors,x∈-r,0×Rand the initial perturbation is u0(s,x)-ϕ(x+cs)∈C[-r,0];Lω1(R)∩H1(R), then the solution of (105) satisfies (110)0=u-≤ut,x≤u+=K,fort,x∈0,+∞×R,ut,x-ϕx+ct∈C0,+∞;Lω1R∩H1R. When c>c∗+B, the solution u(t,x) converges to the traveling wave ϕ(x+ct) exponentially (111)supx∈R⁡ut,x-ϕx+ct≤Ce-μt,t≥0,with a positive constant μ=μ1/3, where μ1 is determined by (112)c-Bλ-Dmλ2+dm-μ1-εb′0e-λcre4B2+4Bλ+λ2-4Br2/16α>0,dm-μ1-εb′Keμ1r>0.Remark 18.
The model can be used to study the behaviour of the mature population. For example, if we take birth functionb(u)=pue-auq, where p,a,q are positive constants and let 1<εp/dm<e1/q, it is easy to verify that, for c>c∗+B, all noncritical traveling waves connecting 0 and 1/aln⁡εp/dm1/q are exponential stable. That is, the density of the population tends to a constant 1/aln⁡εp/dm1/q as time becomes large. The results show that the density of the population will eventually reach a stable state.



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*Source: 4614925-2017-11-06.xml*