Global stability of traveling waves for a spatially discrete diffusion system with time delay

  • Received: 01 September 2020 Revised: 01 November 2020 Published: 11 January 2021
  • Primary: 35K57, 35B35; Secondary: 92D30

  • This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in L(R)×L(R) with the exponential convergence rate eμt for some constant μ>0.

    Citation: Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay[J]. Electronic Research Archive, 2021, 29(4): 2599-2618. doi: 10.3934/era.2021003

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  • This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in L(R)×L(R) with the exponential convergence rate eμt for some constant μ>0.



    In this article, we consider the following spatially discrete diffusion system with time delay

    {tv1(x,t)=d1D[v1](x,t)αv1(x,t)+h(v2(x,tτ1)),tv2(x,t)=d2D[v2](x,t)βv2(x,t)+g(v1(x,tτ2)) (1)

    with the initial data

    vi(x,s)=vi0(x,s), xR, s[τi,0], i=1,2, (2)

    where t>0, xR, di0 and

    D[vi](x,t)=vi(x+1,t)2vi(x,t)+vi(x1,t), i=1,2.

    Here v1(x,t) and v2(x,t) biologically stand for the spatial density of the bacterial population and the infective human population at point xR and time t0, respectively. Both bacteria and humans are assumed to diffuse, d1 and d2 are diffusion coefficients; the term αv1 is the natural death rate of the bacterial population and the nonlinearity h(v2) is the contribution of the infective humans to the growth rate of the bacterial; βv2 is the natural diminishing rate of the infective population due to the finite mean duration of the infectious population and the nonlinearity g(v1) is the infection rate of the human population under the assumption that the total susceptible human population is constant during the evolution of the epidemic, and τ1, τ2 are time delays. The nonlinearities g and h satisfy the following hypothesis:

    (H1) gC2([0,K1],R), g(0)=h(0)=0, K2=g(K1)/β>0, hC2([0,K2],R), h(g(K1)/β)=αK1, h(g(v)/β)>αv for v(0,K1), where K1 is a positive constant.

    According to (H1), the spatially homogeneous system of (1) admits two constant equilibria

    (v1,v2)=0:=(0,0)and(v1+,v2+)=K:=(K1,K2).

    It is clear that (H1) is a basic assumption to ensure that system (1) is monostable on [0,K]. When g(u)0 for u[0,K1] and h(v)0 for v[0,K2], system (1) is a quasi-monotone system. Otherwise, if g(u)0 for u[0,K1] or h(v)0 for v[0,K2] does not hold, system (1) is a non-quasi-monotone system. In this article, we are interested in the existence and stability of traveling wave solutions of (1) connecting two constant equilibria (0,0) and (K1,K2). A traveling wave solution (in short, traveling wave) of (1) is a special translation invariant solution of the form (v1(x,t),v2(x,t))=(ϕ1(x+ct),ϕ2(x+ct)), where c>0 is the wave speed. If ϕ1 and ϕ2 are monotone, then (ϕ1,ϕ2) is called a traveling wavefront. Substituting (ϕ1(x+ct),ϕ2(x+ct)) into (1), we obtain the following wave profile system with the boundary conditions

    {cϕ1(ξ)=d1D[ϕ1](ξ)αϕ1(ξ)+h(ϕ2(ξcτ1)),cϕ2(ξ)=d2D[ϕ2](ξ)βϕ2(ξ)+g(ϕ1(ξcτ2)),(ϕ1,ϕ2)()=(v1,v2),(ϕ1,ϕ2)(+)=(v1+,v2+), (3)

    where ξ=x+ct, =ddξ, D[ϕi](ξ)=ϕi(ξ+1)2ϕi(ξ)+ϕi(ξ1), i=1,2.

    System (1) is a discrete version of classical epidemic model

    {tv1(x,t)=d1xxv1(x,t)a1v1(x,t)+h(v2(x,tτ1)),tv2(x,t)=d2xxv2(x,t)a2v2(x,t)+g(v1(x,tτ2)). (4)

    The existence and stability of traveling waves of (4) have been extensively studied, see [7,19,21,24] and references therein. Note that system (1) is also a delay version of the following system

    {tv1(x,t)=d1D[v1](x,t)a1v1(x,t)+h(v2(x,t)),tv2(x,t)=d2D[v2](x,t)a2v2(x,t)+g(v1(x,t)). (5)

    When system (5) is a quasi-monotone system, Yu, Wan and Hsu [27] established the existence and stability of traveling waves of (5). To the best of our knowledge, when systems (1) and (5) are non-quasi-monotone systems, no result on the existence and stability of traveling waves has been reported. We should point out that the existence of traveling waves of (1) can be easily obtained. Hence, the main purpose of the current paper is to establish the stability of traveling waves of (1).

    The stability of traveling waves for the classical reaction-diffusion equations with and without time delay has been extensively investigated, see e.g., [4,9,10,12,13,14,16,22,24]. Compared to the rich results for the classical reaction-diffusion equations, limited results exist for the spatial discrete diffusion equations. Chen and Guo [1] took the squeezing technique to prove the asymptotic stability of traveling waves for discrete quasilinear monostable equations without time delay. Guo and Zimmer [5] proved the global stability of traveling wavefronts for spatially discrete equations with nonlocal delay effects by using a combination of the weighted energy method and the Green function technique. Tian and Zhang [19] investigated the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with two species by the weighted energy method together with the comparison principle. Later on, Chen, Wu and Hsu [2] employed the similar method to show the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with three species. We should point out that the methods for the above stability results heavily depend on the monotonicity of equations and the comparison principle. However, the most interesting cases are the equations without monotonicity. It is known that when the evolution equations are non-monotone, the comparison principle is not applicable. Thus, the methods, such as the squeezing technique, the weighted energy method combining with the comparison principle are not valid for the stability of traveling waves of the spatial discrete diffusion equations without monotonicity.

    Recently, the technical weighted energy method without the comparison principle has been used to prove the stability of traveling waves of nonmonotone equations, see Chern et al. [3], Lin et al. [10], Wu et al. [22], Yang et al. [24]. In particular, Tian et al. [20] and Yang et al. [26], respectively, applied this method to prove the local stability of traveling waves for nonmonotone traveling waves for spatially discrete reaction-diffusion equations with time delay. Later, Yang and Zhang [25] established the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Unfortunately, the local stability (the initial perturbation around the traveling wave is properly small in a weighted norm) of traveling waves can only be obtained. Very recently, Mei et al. [15] developed a new method to prove the global stability of the oscillatory traveling waves of local Nicholson's blowflies equations. This method is based on some key observations for the structure of the govern equations and the anti-weighted energy method together with the Fourier transform. Later on, Zhang [28] and Xu et al. [23], respectively, applied this method successfully to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. More recently, Su and Zhang [17] further studied a discrete diffusion equation with a monostable convolution type nonlinearity and established the global stability of traveling waves with large speed. Motivated by the works [15,28,23,17,18], in this paper, we shall extend this method to study the global stability of traveling waves of spatial discrete diffusion system (1) without quasi-monotonicity.

    The rest of this paper is organized as follows. In Section 2, we present some preliminaries and summarize our main results. Section 3 is dedicated to the global stability of traveling waves of (1) by the Fourier transform and the weighted energy method, when h(u) and g(u) are not monotone.

    In this section, we first give the equivalent integral form of the initial value problem of (1) with (2), then recall the existence of traveling waves of (1), and finally state the main result on the global stability of traveling waves of (1). Throughout this paper, we assume τ1=τ2=τ.

    First of all, we consider the initial value problem (1) with (2), i.e.,

    {tv1(x,t)=d1D[v1](x,t)αv1(x,t)+h(v2(x,tτ)),tv2(x,t)=d2D[v2](x,t)βv2(x,t)+g(v1(x,tτ)),vi(x,s)=vi0(x,s), xR, s[τ,0], i=1,2. (6)

    According to [8], with aid of modified Bessel functions, the solution to the initial value problem

    {tu(x,t)=d[u(x+1,t)2u(x,t)+u(x1,t)], xR, t>0,u(x,0)=u0(x), xR,

    can be expressed by

    u(x,t)=(S(t)u0)(x)=e2dtm=Im(2dt)u0(xm),

    where u0()L(R), Im(), m0 are defined as

    Im(t)=k=0(t/2)m+2kk!(m+k)!,

    and Im(t)=Im(t) for m<0. Moreover,

    Im(t)=12[Im+1(t)+Im1(t)], t>0,mZ, (7)

    and Im(0)=0 for m0 while I0=1, and Im(t)0 for any t0. In addition, one has

    etm=Im(t)=et[I0(t)+2I1(t)+2I2(t)+I3(t)+]=1. (8)

    Thus, the solution (v1(x,t),v2(x,t)) of (6) can be expressed as

    {v1(x,t)=e(2d1+α)tm=Im(2d1t)v10(xm,0)+m=0te(2d1+α)(ts)Im(2d1(ts))(h(v2(xm,sτ)))ds,v2(x,t)=e(2d2+β)tm=Im(2d2t)v20(xm,0)+m=0te(2d2+β)(ts)Im(2d2(ts))(g(v1(xm,sτ)))ds. (9)

    In fact, by [8,Lemma 2.1], we can differentiate the series on t variable in (9). Using the recurrence relation (7), we obtain

    tv1(x,t)=(2d1+α)e(2d1+α)tm=Im(2d1t)v10(xm,0)   +e(2d1+α)tm=2d1Im(2d1t)v10(xm,0)   +m=Im(0)(h(v2(xm,tτ)))   (2d1+α)m=0te(2d1+α)(ts)Im(2d1(ts))(h(v2(xm,sτ)))ds   +m=0te(2d1+α)(ts)2d1Im(2d1(ts))(h(v2(xm,sτ)))ds=d1[v1(x+1,t)2v1(x,t)+v1(x1,t)]αv1(x,t)+h(v2(x,tτ))

    and

    tv2(x,t)=(2d2+β)e(2d2+β)tm=Im(2d2t)v20(xm,0)   +e(2d2+β)tm=2d2Im(2d2t)v20(xm,0)   +m=Im(0)(g(v1(xm,tτ)))   (2d2+β)m=0te(2d2+β)(ts)Im(2d2(ts))(g(v1(xm,sτ)))ds   +m=0te(2d2+β)(ts)2d2Im(2d2(ts))(g(v1(xm,sτ)))ds=d2[v2(x+1,t)2v2(x,t)+v2(x1,t)]βv2(x,t)+g(v1(x,tτ)).

    Next we investigate the characteristic roots of the linearized system for the wave profile system (3) at the trivial equilibrium 0. Clearly, the characteristic function of (3) at 0 is

    P1(c,λ):=f1(c,λ)f2(c,λ)

    for c0 and λC, where

    f1(c,λ):=Δ1(c,λ)Δ2(c,λ),f2(c,λ):=h(0)g(0)e2cλτ,

    with

    Δ1(c,λ)=d1(eλ+eλ2)cλα,Δ2(c,λ)=d2(eλ+eλ2)cλβ.

    It is easy to see that Δ1(c,λ)=0 admits two roots λ1<0<λ1+, and Δ2(c,λ)=0 has two roots λ2<0<λ2+. We denote λm+=min{λ1+,λ2+}.

    Similar to [27,Lemma 3.1], we can obtain the following result.

    Lemma 2.1. There exists a positive constant c such that if c>c, then P1(c,λ)=0 has two distinct positive real roots λ1:=λ1(c) and λ2:=λ2(c) with λ1(c)<λ2(c)<λm+, i.e. P1(c,λ1)=P1(c,λ2)=0, and P(c,λ)>0 for λ(λ1(c),λ2(c)). In addition, limccλ1(c)=limccλ2(c)=λ>0, i.e., P1(c,λ)=0.

    Furthermore, we show the existence of traveling wave of (1). When system (1) is a quasi-monotone system, the existence of traveling wavefronts follows from [6,Theorem 1.1]. When system (1) is a non-quasi-monotone system, the existence of traveling waves can also be obtained by using auxiliary equations and Schauder's fixed point theorem [21,24], if we assume the following assumptions:

    (H2) There exist K±=(K1±,K2±)0 with K<K<K+ and four continuous and twice piecewise continuous differentiable functions g±:[0,K1+]R and h±:[0,K2+]R such that

    (i) K2±=g±(K1±)/β, h±(1βg±(K1±))=αK1±, and h±(1βg±(v))>αv for v(0,K1±);

    (ii) g±(u) and h±(v) are non-decreasing on [0,K1+] and [0,K2+], respectively;

    (iii) (g±)(0)=g(0), (h±)(0)=h(0) and

    0<g(u)g(u)g+(u)g(0)u for u[0,K1+],0<h(v)h(v)h+(v)h(0)v for v[0,K2+].

    Proposition 1. Assume that (H1) and (H2) hold, τ0, and let c be defined as in Lemma 2.1. Then for every c>c, system (1) has a traveling wave (ϕ1(ξ),ϕ2(ξ)) satisfying (ϕ1(),ϕ2())=(0,0) and

    K1lim infξ+ϕ1(ξ)lim supξ+ϕ1(ξ)K1+,0lim infξ+ϕ2(ξ)lim supξ+ϕ2(ξ)K2+.

    Finally, we shall state the stability result of traveling waves derived in Proposition 1. Before that, let us introduce the following notations.

    Notations. C>0 denotes a generic constant, while Ci(i=1,2,) represents a specific constant. Let and denote 1-norm and -norm of the matrix (or vector), respectively. Let I be an interval, typically I=R. Denote by L1(I) the space of integrable functions defined on I, and Wk,1(I)(k0) the Sobolev space of the L1-functions f(x) defined on the interval I whose derivatives dndxnf(n=1,,k) also belong to L1(I). Let Lw1(I) be the weighted L1-space with a weight function w(x)>0 and its norm is defined by

    ||f||Lw1(I)=Iw(x)|f(x)|dx,

    Wwk,1(I) be the weighted Sobolev space with the norm given by

    ||f||Wwk,1(I)=i=0kIw(x)|dif(x)dxi|dx.

    Let T>0 be a number and B be a Banach space. We denote by C([0,T];B) the space of the B-valued continuous functions on [0,T], and by L1([0,T];B) the space of the B-valued L1-functions on [0,T]. The corresponding spaces of the B-valued functions on [0,) are defined similarly. For any function f(x), its Fourier transform is defined by

    F[f](η)=f^(η)=Reixηf(x)dx

    and the inverse Fourier transform is given by

    F1[f^](x)=12πReixηf^(η)dη,

    where i is the imaginary unit, i2=1.

    To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.

    (H3) |g(u)|g(0) and |h(v)|h(0) for u,v[0,+).

    (H4) d2>d1, α>β, d2d1<αβ2 and max{h(0),g(0)}>β.

    (H5) The initial data (v10(x,s),v20(x,s))(0,0) satisfies

    limx±(v10(x,s),v20(x,s))=(v1±,v2±) uniformly in  s[τ,0].

    Consider the following function

    P2(λ,c)=d2(eλ+eλ2)cλβ+max{h(0),g(0)}eλcτ.

    Since max{h(0),g(0)}>β, it then follows from [20,Lemma 2.1] that there exists λ>0 and c>0, such that P2(λ,c)=0 and P2(λ,c)λ|(λ,c)=0. When c>c, the equation P2(λ,c)=0 has two positive real roots λ1(c) and λ2(c) with 0<λ1(c)<λ<λ2(c). When λ(λ1(c),λ2(c)), P2(λ,c)<0. Moreover, (λ1)(c)<0 and (λ2)(c)>0.

    We select the weight function w(ξ)>0 as the form

    w(ξ)=e2λξ,

    where λ>0 satisfies λ1(c)<λ<λ2(c). Now we are ready to present the main result of this paper.

    Theorem 2.2. (Global stability of traveling waves). Assume that (H1), (H3)-(H5) hold. For any given traveling wave (ϕ1(x+ct),ϕ2(x+ct)) of (1) with speed c>max{c,c} connecting (0,0) and (K1,K2), whether it is monotone or non-monotone, if the initial data satisfy

    vi0(x,s)ϕi(x+cs)Cunif[τ,0]C([τ,0];Ww1,1(R)), i=1,2,s(vi0ϕi)L1([τ,0];Lw1(R)), i=1,2,

    then there exists τ0>0 such that for any ττ0, the solution (v1(x,t),v2(x,t)) of (1)-(2) converges to the traveling wave (ϕ1(x+ct),ϕ2(x+ct)) as follows:

    supxR|vi(x,t)ϕi(x+ct)|Ceμt,t>0,

    where C and μ are two positive constants, and Cunif[r,T] is the uniformly continuous space, for 0<T, defined by

    Cunif[r,T]={uC([r,T]×R)such thatlimx+v(x,t)exists uniformly int[r,T]}.

    This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let (ϕ1(x+ct),ϕ2(x+ct))=(ϕ1(ξ),ϕ2(ξ)) be a given traveling wave solution with speed cc and define

    {Vi(ξ,t):=vi(x,t)ϕi(x+ct)=vi(ξct,t)ϕi(ξ), i=1,2,Vi0(ξ,s):=vi0(x,s)ϕi(x+cs)=vi0(ξcs,s)ϕ(ξ), i=1,2.

    Then it follows from (1) and (3) that Vi(ξ,t) satisfies

    {V1t+cV1ξd1D[V1]+αV1=Q1(V2(ξcτ,tτ)),V2t+cV2ξd2D[V2]+βV2=Q2(V1(ξcτ,tτ)),Vi(ξ,s)=Vi0(ξ,s), (ξ,s)R×[τ,0], i=1,2. (10)

    The nonlinear terms Q1 and Q2 are given by

    {Q1(V2):=h(ϕ2+V2)h(ϕ2)=h(ϕ~2)V2,Q2(V1):=g(ϕ1+V1)g(ϕ1)=g(ϕ~1)V1, (11)

    for some ϕ~i between ϕi and ϕi+Vi, with ϕi=ϕi(ξcτi) and Vi=Vi(ξcτi,tτi).

    We first prove the existence and uniqueness of solution (V1(ξ,t),V2(ξ,t)) to the initial value problem (10) in the uniformly continuous space Cunif[τ,+)×Cunif[τ,+).

    Lemma 3.1. Assume that (H1)and(H3) hold. If the initial perturbation (V10,V20)Cunif[τ,0]×Cunif[τ,0] for cc, then the solution (V1,V2) of the perturbed equation (10) is unique and time-globally exists in Cunif[τ,+)×Cunif[τ,+).

    Proof. Let Ui(x,t)=vi(x,t)ϕi(x+ct), i=1,2. It is clear that Ui(x,t)=Vi(ξ,t), i=1,2, and satisfies

    {U1td1D[U1]+αU1=Q1(U2(x,tτ)),U2td2D[U2]+βU2=Q2(U1(x,tτ)),Ui(x,s)=vi0(x,s)ϕi(x+cs):=Ui0(x,s), (x,s)R×[τ,0], i=1,2. (12)

    Thus, the global existence and uniqueness of solution of (10) are transformed into that of (12).

    When t[0,τ], we have tτ[τ,0] and Ui(x,tτ)=Ui0(x,tτ), i=1,2, which imply that (12) is linear. Thus, the solution of (12) can be explicitly and uniquely solved by

    {U1(x,t)=e(2d1+α)tm=Im(2d1t)U10(xm,0)             +m=0te(2d1+α)(ts)Im(2d1(ts))Q1(U20(xm,sτ))ds,U2(x,t)=e(2d2+β)tm=Im(2d2t)U20(xm,0)             +m=0te(2d2+β)(ts)Im(2d2(ts))Q2(U10(xm,sτ))ds (13)

    for t[0,τ].

    Since Vi0(ξ,t)Cunif[τ,0], i=1,2, namely, limξ+Vi0(ξ,t) exist uniformly in t[τ,0], which implies limx+Ui0(x,t) exist uniformly in t[τ,0]. Denote Ui0(,t)=limx+Ui0(x,t), i=1,2. Taking the limit x+ to (13) yields

    limx+U1(x,t)=e(2d1+α)tm=Im(2d1t)limx+U10(xm,0)+m=0te(2d1+α)(ts)Im(2d1(ts))limx+Q1(U20(xm,sτ))ds=eαtU10(,0)+0teα(ts)Q1(U20(,sτ))m=e2d1(ts)Im(2d1(ts))ds=:U1(t)  uniformly in t[0,τ] (14)

    and

    limx+U2(x,t)=e(2d2+β)tm=Im(2d2t)limx+U20(xm,0)
    +m=0te(2d2+β)(ts)Im(2d2(ts))limx+Q2(U10(xm,sτ))ds=eβtU20(,0)+0teβ(ts)Q2(U10(,sτ))m=e2d2(ts)Im(2d2(ts))ds=:U2(t)  uniformly in t[0,τ], (15)

    where we have used (8). Thus, we obtain that (U1,U2)Cunif[τ,τ)×Cunif[τ,τ).

    When t[τ,2τ], system (12) with the initial data Ui(x,s) for s[0,τ] is still linear, because the source term Q1(U2(x,tτ)) and Q2(U1(x,tτ)) is known due to tτ[0,τ] and Ui(s,tτ) is solved in (13). Hence, the solution Ui(x,t) for t[τ,2τ] is uniquely and explicitly given by

    U1(x,t)=e(2d1+α)(tτ)m=Im(2d1(tτ))U1(xm,τ)+m=τte(2d1+α)(ts)Im(2d1(ts))Q1(U2(xm,sτ))ds,U2(x,t)=e(2d2+β)(tτ)m=Im(2d2(tτ))U2(xm,τ)+m=τte(2d2+β)(ts)Im(2d2(ts))Q2(U1(xm,sτ))ds.

    Similarly, by (14) and (15), we have

    limx+U1(x,t)=e(2d1+α)(tτ)m=Im(2d1(tτ))limx+U1(xm,τ)+m=τte(2d1+α)(ts)Im(2d1(ts))limx+Q1(U2(xm,sτ))ds=eα(tτ)U1(τ)+τteα(ts)Q1(U1(sτ))m=e2d1(ts)Im(2d1(ts))ds=:U¯1(t)  uniformly in t[τ,2τ],

    and

    limx+U2(x,t)=e(2d2+β)(tτ)m=Im(2d2(tτ))limx+U2(xm,τ)+m=τte(2d2+β)(ts)Im(2d2(ts))limx+Q2(U1(xm,sτ))ds=eβ(tτ)U2(τ)+τteβ(ts)Q2(U2(sτ))m=e2d2(ts)Im(2d2(ts))ds
    =:U¯2(t)  uniformly in t[τ,2τ].

    By repeating this procedure for t[nτ,(n+1)τ] with nZ+, we prove that there exists a unique solution (V1,V2)Cunif[τ,(n+1)τ]×Cunif[τ,(n+1)τ] for (10), and step by step, we finally prove the uniqueness and time-global existence of the solution (V1,V2)Cunif[τ,)×Cunif[τ,) for (10). The proof is complete.

    Now we state the stability result for the perturbed system (10), which automatically implies Theorem 2.2.

    Proposition 2. Assume that (H1), (H3)-(H5) hold. If

    Vi0Cunif[τ,0]C([τ,0];Ww1,1(R)), i=1,2,

    and

    sVi0L1([τ,0];Lw1(R)), i=1,2,

    then there exists τ0>0 such that for any ττ0, when c>max{c,c}, it holds

    supξR|Vi(ξ,t)|Ceμt,t>0, i=1,2, (16)

    for some μ>0 and C>0.

    In order to prove Proposition 2, we first investigate the decay estimate of Vi(ξ,t) at ξ=+, i=1,2.

    Lemma 3.2. Assume that Vi0Cunif[τ,0], i=1,2. Then, there exist τ0>0 and a large number x01 such that when ττ0, the solution Vi(ξ,t) of (10) satisfies

    supξ[x0,+)|Vi(ξ,t)|Ceμ1t, t>0, i=1,2,

    for some μ1>0 and C>0.

    Proof. Denote

    zi+(t):=Vi(,t), zi0+(s):=Vi0(,s), s[τ,0], i=1,2.

    Since Vi0Cunif[τ,0], i=1,2, by Lemma 3.1, we have ViCunif[τ,+), which implies

    limξ+Vi(ξ,t)=zi+(t)

    exists uniformly for t[τ,+). Taking the limit ξ+ to (10), we obtain

    {dz1+dt+αz1+h(v2+)z2+(tτ)=P1(z2+(tτ)),dz2+dt+βz2+g(v1+)z1+(tτ)=P2(z1+(tτ)),zi+(s)=zi0+(s), s[τ,0], i=1,2,

    where

    {P1(z2+)=h(v2++z2+)h(v2+)h(v2+)z2+,P2(z1+)=g(v1++z1+)g(v1+)g(v1+)z1+.

    Then by [9,Lemma 3.8], there exist positive constants τ0, μ1 and C such that when ττ0,

    |Vi(,t)|=|zi+(t)|Ceμ1t, t>0, i=1,2, (17)

    provided that |zi0+|1, i=1,2.

    By the continuity and the uniform convergence of Vi(ξ,t) as ξ+, there exists a large x01 such that (17) implies

    supξ[x0,+)|Vi(ξ,t)|Ceμ1t, t>0, i=1,2,

    provided that supξ[x0,+)|Vi0(ξ,s)|1 for s[τ,0]. Such a smallness for the initial perturbation (V10,V20) near ξ+ can be easily verified, since

    limx+(v10(x,s),v20(x,s))=(K1,K2) uniformly in s[τ,0],

    which implies

    limξ+Vi0(ξ,s)=limξ+[vi0(ξ,s)ϕi(ξ)]=KiKi=0

    uniformly for s[τ,0], i=1,2. The proof is complete.

    Next we are going to establish the a priori decay estimate of supξ(,x0]|Vi(ξ,t)| by using the anti-weighted technique [3] together with the Fourier transform. First of all, we shift Vi(ξ,t) to Vi(ξ+x0,t) by the constant x0 given in Lemma 3.2, and then introduce the following transformation

    V~i(ξ,t)=w(ξ)Vi(ξ+x0,t)=eλξVi(ξ+x0,t),i=1,2.

    Substituting Vi=w1/2V~i to (10) yields

    {V~1t+cV~1ξ+c1V~1(ξ,t)d1eλV~1(ξ+1,t)d1eλV~1(ξ1,t)=Q~1(V~2(ξcτ,tτ)),V~2t+cV~2ξ+c2V~2(ξ,t)d2eλV~2(ξ+1,t)d2eλV~2(ξ1,t)=Q~2(V~1(ξcτ,tτ)),V~i(ξ,s)=w(ξ)Vi0(ξ+x0,s)=:V~i0(ξ,s), ξR,s[τ,0], i=1,2, (18)

    where

    c1=cλ+2d1+α,c2=cλ+2d2+β

    and

    Q~1(V~2)=eλξQ1(V2),Q~2(V~1)=eλξQ2(V1).

    By (11), Q~1(V~2) satisfies

    Q~1(V~2(ξcτ,tτ))=eλξQ1(V2(ξcτ+x0,tτ))=eλξh(ϕ~2)V2(ξcτ+x0,tτ)=eλcτh(ϕ~2)V~2(ξcτ,tτ) (19)

    and Q~2(V~1) satisfies

    Q~2(V~1(ξcτ,tτ))=eλcτg(ϕ~1)V~1(ξcτ,tτ). (20)

    By (H3), we further obtain

    |Q~1(V~2(ξcτ,tτ))|h(0)eλcτ|V~2(ξcτ,tτ)|,|Q~2(V~1(ξcτ,tτ))|g(0)eλcτ|V~1(ξcτ,tτ)|.

    Taking (19) and (20) into (18), one can see that the coefficients h(ϕ~2) and g(ϕ~1) on the right side of (18) are variable and can be negative. Thus, the classical methods, such as the monotone technique and the Fourier transform cannot be applied directly to establish the decay estimate for (V~1,V~2). Motivated by [15,28,17,23], we introduce a new method which can be described as follows.

    By replacing h(ϕ~2) in the first equation of (18) with a constant h(0), and g(ϕ~1) in the second equation of (18) with a constant g(0), we can obtain a linear delayed reaction-diffusion system

    {V1t++cV1ξ++c1V1+(ξ,t)d1eλV1+(ξ+1,t)d1eλV1+(ξ1,t) =h(0)eλcτV2+(ξcτ,tτ),V2t++cV2ξ++c2V2+(ξ,t)d2eλV2+(ξ+1,t)d2eλV2+(ξ1,t) =g(0)eλcτV1+(ξcτ,tτ), (21)

    with

    Vi+(ξ,s)=w(ξ)Vi0(ξ+x0,s)=:Vi0+(ξ,s), i=1,2,

    where ξR, t(0,+] and s[τ,0]. Then we investigate the decay estimate of (V1+,V2+) by applying the Fourier transform to (21);

    We prove that the solution (V~1,V~2) of (18) can be bounded by the solution (V1+,V2+) of (21).

    Now we are in a position to derive the decay estimate of (V1+,V2+) for the linear system (21). We first recall some properties of the solutions to the delayed ODE system.

    Lemma 3.3. ([11,Lemma 3.1]) Let z(t) be the solution to the following scalar differential equation with delay

    {ddtz(t)=Az(t)+Bz(tτ),t0,τ>0,z(s)=z0(s),s[τ,0]. (22)

    where A,BCN×N, N2, and z0(s)C1([τ,0],CN). Then

    z(t)=eA(t+τ)eτB1tz0(τ)+τ0eA(ts)eτB1(tτs)[z0(s)Az0(s)]ds,

    where B1=BeAτ and eτB1t is the so-called delayed exponential function in the form

    eτB1t={0,<t<τ,I,τt<0,I+B1t1!,0t<τ,I+B1t1!+B12(tτ)22!,τt<2τ,I+B1t1!+B12(tτ)22!++B1m[t(m1)τ]mm!,(m1)τt<mτ,

    where 0,ICN×N, and 0 is zero matrix and I is unit matrix.

    Lemma 3.4. ([11,Theorem 3.1]) Suppose μ(A):=μ1(A)+μ(A)2<0, where μ1(A) and μ(A) denote the matrix measure of A induced by the matrix 1-norm 1 and -norm , respectively. If ν(B):=B+B2μ(A), then there exists a decreasing function ετ=ε(τ)(0,1) for τ>0 such that any solution of system (22) satisfies

    z(t)C0eετσt,t>0,

    where C0 is a positive constant depending on initial data z0(s),s[τ,0] and σ=|μ(A)|ν(B). In particular,

    eAteτB1tC0eετσt,t>0,

    where eτB1t is defined in Lemma 3.3.

    From the proof of [11,Theome 3.1], one can see that

    μ1(A)=limθ0+I+θA1θ=max1jN[Re(ajj)+jiN|aij|]

    and

    μ(A)=limθ0+I+θA1θ=max1iN[Re(aii)+ijN|aij|].

    Taking the Fourier transform to (21) and denoting the Fourier transform of V+(ξ,t):=(V1+(ξ,t),V2+(ξ,t))T by V^+(η,t):=(V^1+(η,t),V^2+(η,t))T, we obtain

    {tV^1+(η,t)=(c1+d1(eλ+iη+e(λ+iη))icη)V^1+(η,t)                 +h(0)ecτ(λ+iη)V^2+(η,tτ),tV^2+(η,t)=(c2+d2(eλ+iη+e(λ+iη))icη)V^2+(η,t)                 +g(0)ecτ(λ+iη)V^1+(η,tτ),V^i+(η,s)=V^i0+(η,s), ηR, s[τ,0], i=1,2. (23)

    Let

    A(η)=(c1+d1(eλ+iη+e(λ+iη))icη00c2+d2(eλ+iη+e(λ+iη))icη)

    and

    B(η)=(0h(0)ecτ(λ+iη)g(0)ecτ(λ+iη)0).

    Then system (23) can be rewritten as

    V^t+(η,t)=A(η)V^+(η,t)+B(η)V^+(η,tτ). (24)

    By Lemma 3.3, the linear delayed system (24) can be solved by

    V^+(η,t)=eA(η)(t+τ)eτB1(η)tV^0+(η,τ)+τ0eA(η)(ts)eτB1(η)(tsτ)[sV^0+(η,s)A(η)V^0+(η,s)]ds:=I1(η,t)+τ0I2(η,ts)ds, (25)

    where B1(η)=B(η)eA(η)τ. Then by taking the inverse Fourier transform to (25), one has

    V+(ξ,t) (26)
    =F1[I1](ξ,t)+τ0F1[I2](ξ,ts)ds=12πeiξηeA(η)(t+τ)eτB1(η)tV^0+(η,τ)dη   +12πτ0eiξηeA(η)(ts)eτB1(η)(tsτ)[sV^0+(η,s)A(η)V^0+(η,s)]dηds. (27)

    Lemma 3.5. Let the initial data Vi0+(ξ,s), i=1,2, be such that

    Vi0+C([τ,0];W1,1(R)), sVi0+L1([τ,0];L1(R)), i=1,2.

    Then

    Vi+(t)L(R)Ceμ2t for cmax{c,c}, i=1,2,

    where μ2>0 and C>0.

    Proof. According to (26), we shall estimate F1[I1](ξ,t) and τ0F1[I2](ξ,ts)ds, respectively. By the definition of μ() and ν(), we have

    μ(A(η))=μ1(A(η))+μ(A(η))2=max{c1+d1(eλcosη+eλcosη),c2+d2(eλcosη+eλcosη)}=c2+d2(eλcosη+eλcosη)=c2+d2(eλ+eλ)cosη=cλ+d2(eλ+eλ2)βm(η),

    where c2=cλ+2d2+β and

    m(η)=d2(1cosη)(eλ+eλ)0,

    since d2>d1, α>β and d2d1<αβ2, and

    ν(B(η))=max{h(0),g(0)}eλcτ.

    By considering λ(λ1(c),λ2(c)), we get μ(A(η))<0 and

    μ(A(η))+ν(B(η))=cλ+d2(eλ+eλ2)βm(η)+max{h(0),g(0)}eλcτ<0.

    Furthermore, we obtain

    |μ(A(η))|ν(B(η))=cλd2(eλ+eλ2)+β+m(η)max{h(0),g(0)}eλcτ=P2(λ,c)+m(η),

    where P2(λ,c)=d2(eλ+eλ2)cλβ+max{h(0),g(0)}eλcτ<0 for c>max{c,c}. It then follows from Lemma 3.4 that there exists a decreasing function ετ=ε(τ)(0,1) such that

    eA(η)(t+τ)eB1(η)tC1eετ(|μ(A(η))|ν(B(η)))tC1eετμ0teετm(η)t, (28)

    where C1 is a positive constant and μ0:=P2(λ,c)>0 with c>c. By the definition of Fourier's transform, we have

    supηRV^0+(η,τ)RV0+(ξ,τ)dξ=i=12Vi0+(,τ)L1(R).

    Applying (28), we derive

    supξRF1[I1](ξ,t)=supξR12πeiξηeA(η)(t+τ)eB1(η)tV^0+(η,τ)dηCeετm(η)teετμ0tV^0+(η,τ)dηCeετμ0tsupηRV^0+(η,τ)eετm(η)tdηCeμ2ti=12Vi0+(,τ)L1(R), (29)

    with μ2:=ετμ0.

    Note that

    supηRA(η)V^0+(η,s)Ci=12Vi0+(,s)W1,1(R).

    Similarly, we can obtain

    supξRF1[I2](ξ,ts)=supξR12πeiξηeA(η)(ts)eB1(η)(tsτ)[sV^0+(η,s)A(η)V^0+(η,s)]dηCeετm(η)(ts)eετμ0(ts)sV^0+(η,s)A(η)V^0+(η,s)dηCeετμ0teετμ0ssupηRsV^0+(η,s)A(η)V^0+(η,s)eετm(η)(ts)dη.

    It then follows that

    τ0supξRF1[I2](ξ,ts)dsCeετμ0tτ0eετμ0ssupηRsV^0+(η,s)A(η)V^0+(η,s)eετm(η)(ts)dηdsCeετμ0tτ0sV0+(,s)L1(R)+V0+(,s)W1,1(R)dsCeετμ0t(sV0+(s)L1([τ,0];L1(R))+V0+(s)L1([τ,0];W1,1(R))). (30)

    Substituting (29) and (30) to (26), we obtain the following the decay rate

    i=12Vi+(t)L(R)Ceμ2t.

    This proof is complete.

    The following maximum principle is needed to obtain the crucial boundedness estimate of (V~1,V~2), which has been proved in [17,Lemma 3.4].

    Lemma 3.6. Let T>0. For any a1,a2R and ν>0, if the bounded function v satisfies

    {vt+a1vξ+a2vdeνv(t,ξ+1)deνv(t,ξ1)0, (t,ξ)(0,T]×R,v(0,ξ)0,ξR, (31)

    then v(t,ξ)0 for all (t,ξ)(0,T]×R.

    Lemma 3.7. When (V10+(ξ,s),V20+(ξ,s))(0,0) for (ξ,s)R×[τ,0], then (V1+(ξ,t),V2+(ξ,t))(0,0) for (ξ,t)R×[0,+).

    Proof. When t[0,τ], we have tτ[τ,0] and

    h(0)eλcτV2+(ξcτ,tτ)=h(0)eλcτV20+(ξcτ,tτ)0. (32)

    Applying (32) to the first equation of (21), we get

    {V1t++cV1ξ++c1V1+(ξ,t)d1eλV1+(ξ+1,t)d1eλV1+(ξ1,t)0, (ξ,t)R×[0,τ],V10+(ξ,s)0, ξR, s[τ,0].

    By Lemma 3.6, we derive

    V1+(ξ,t)0,(ξ,t)R×[0,τ]. (33)

    Similarly, we obtain

    {V2t++cV2ξ++c2V2+(ξ,t)d2eλV2+(ξ+1,t)d2eλV2+(ξ1,t)0, (ξ,t)R×[0,τ],V20+(ξ,s)0, ξR s[τ,0].

    Using Lemma 3.6 again, we obtain

    V2+(ξ,t)0,(ξ,t)R×[0,τ]. (34)

    When t[nτ,(n+1)τ], n=1,2,, repeating the above procedure step by step, we can similarly prove

    (V1+(ξ,t),V2+(ξ,t))(0,0),(ξ,t)R×[nτ,(n+1)τ]. (35)

    Combining (33), (34) and (31), we obtain (V1+(ξ,t),V2+(ξ,t))(0,0) for (ξ,t)R×[0,+). The proof is complete.

    Now we establish the following crucial boundedness estimate for (V~1,V~2).

    Lemma 3.8. Let (V~1(ξ,t),V~2(ξ,t)) and (V1+(ξ,t),V2+(ξ,t)) be the solutions of (18) and (21), respectively. When

    |V~i0(ξ,s)|Vi0+(ξ,s)for(ξ,s)R×[τ,0], i=1,2, (36)

    then

    |V~i(ξ,t)|Vi+(ξ,t)for(ξ,t)R×[0,+), i=1,2.

    Proof. First of all, we prove |V~i(ξ,t)|Vi+(ξ,t) for t[0,τ],i=1,2. In fact, when t[0,τ], namely, tτ[τ,0], it follows from (36) that

    |V~i(ξcτ,tτ)|=|V~i0(ξcτ,tτ)|Vi0+(ξcτ,tτ)=Vi+(ξcτ,tτ)for (ξ,t)R×[0,τ]. (37)

    Then by |h(ϕ~2)|<h(0) and |g(ϕ~1)|<g(0) and (37), we get

    h(0)eλcτV2+(ξcτ,tτ)±h(ϕ~2)eλcτV~2(ξcτ,tτ)h(0)eλcτV2+(ξcτ,tτ)|h(ϕ~2)|eλcτ|V~2(ξcτ,tτ)|0for (ξ,t)R×[0,τ] (38)

    and

    g(0)eλcτV1+(ξcτ,tτ)±g(ϕ~1)eλcτV~1(ξcτ,tτ)0for (ξ,t)R×[0,τ]. (39)

    Let

    Ui(ξ,t):=Vi+(ξ,t)V~i(ξ,t)andUi+(ξ,t):=Vi+(ξ,t)+V~i(ξ,t),i=1,2.

    We are going to estimate Ui±(ξ,t) respectively.

    From (18), (19), (21) and (38), we see that U1(ξ,t) satisfies

    {U1t+cU1ξ+c1U1(ξ,t)d1eλU1(ξ+1,t)d1eλU1(ξ1,t)0,(ξ,t)R×[0,τ],U10(ξ,s)=V10+(ξ,s)V~10(ξ,s)0,ξR, s[τ,0].

    By Lemma 3.6, we obtain

    U1(ξ,t)0,(ξ,t)R×[0,τ],

    namely,

    V~1(ξ,t)V1+(ξ,t),(ξ,t)R×[0,τ]. (40)

    Similarly, one has

    {U2t+cU2ξ+c2U2(ξ,t)d2eλU2(ξ+1,t)d2eλU2(ξ1,t)0,(ξ,t)R×[0,τ],U20(ξ,s)=V20+(ξ,s)V~20(ξ,s)0,ξR, s[τ,0].

    Applying Lemma 3.6 again, we have

    U2(ξ,t)0,(ξ,t)R×[0,τ],

    i.e.,

    V~2(ξ,t)V2+(ξ,t),(ξ,t)R×[0,τ]. (41)

    On the other hand, U1+(ξ,t) satisfies

    {U1t++cU1ξ++c1U1+(ξ,t)d1eλU1+(ξ+1,t)d1eλU1+(ξ1,t)0,(ξ,t)R×[0,τ],U10(ξ,s)=V10+(ξ,s)V~10(ξ,s)0,ξR, s[τ,0].

    Then Lemma 3.6 implies that

    U1+(ξ,t)=V1+(ξ,t)+V~1(ξ,t)0,(ξ,t)R×[0,τ],

    that is,

    V1+(ξ,t)V~1(ξ,t),(ξ,t)R×[0,τ]. (42)

    Similarly, U2+(ξ,t) satisfies

    {U2t++cU2ξ++c2U2+(ξ,t)d2eλU2+(ξ+1,t)d2eλU2+(ξ1,t)0,(ξ,t)R×[0,τ],U20(ξ,s)=V20+(ξ,s)V~10(ξ,s)0,ξR, s[τ,0].

    Therefore, we can prove that

    U2+(ξ,t)=V2+(ξ,t)+V~2(ξ,t)0,(ξ,t)R×[0,τ],

    namely

    V2+(ξ,t)V~2(ξ,t),(ξ,t)R×[0,τ]. (43)

    Combining (40) and (42), we obtain

    |V~1(ξ,t)|V1+(ξ,t)for(ξ,t)R×[0,τ], (44)

    and combining (41) and (43), we prove

    |V~2(ξ,t)|V2+(ξ,t)for(ξ,t)R×[0,τ]. (45)

    Next, when t[τ,2τ], namely, tτ[0,τ], based on (44) and (45), we can similarly prove

    |V~i(ξ,t)|Vi+(ξ,t)for(ξ,t)R×[τ,2τ],i=1,2.

    Repeating this procedure, we then further prove

    |V~i(ξ,t)|Vi+(ξ,t),(ξ,t)R×[nτ,(n+1)τ],n=1,2,,

    which implies

    |V~i(ξ,t)|Vi+(ξ,t)for(ξ,t)R×[0,),i=1,2.

    The proof is complete.

    Let us choose Vi0+(ξ,s) such that

    Vi0+C([τ,0];W1,1(R)),sVi0+L1([τ,0];L1(R)),

    and

    Vi0+(ξ,s)|Vi0(ξ,s)|,(ξ,s)R×[τ,0], i=1,2.

    Combining Lemmas 3.5 and 3.8, we can get the convergence rates for V~(ξ,t).

    Lemma 3.9. When V~i0C([τ,0];W1,1(R)) and sV~i0L1([τ,0];L1(R)), then

    V~i(t)L(R)Ceμ2t,

    for some μ2>0, i=1,2.

    Lemma 3.10. It holds that

    supξ(,x0]|Vi(ξ,t)|Ceμ2t, i=1,2,

    for some μ2>0.

    Proof. Since V~i(ξ,t)=w(ξ)Vi(ξ+x0,t)=eλξVi(ξ+x0,t) and w(ξ)=eλξ1 for ξ(,0], then we obtain

    supξ(,0]|Vi(ξ+x0,t)|V~i(t)L(R)Ceμ2t,

    which implies

    supξ(,x0]|Vi(ξ,t)|Ceμ2t.

    Thus, the estimate for the unshifted V(ξ,t) is obtained. The proof is complete.

    Proof of Proposition 3.2. By Lemmas 3.2 and 3.10, we immediately obtain (16) for 0<μ<min{μ1,μ2}.

    We are grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.



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