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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,t−τ1)),∂tv2(x,t)=d2D[v2](x,t)−βv2(x,t)+g(v1(x,t−τ2)) $ | (1) |
with the initial data
$ vi(x,s)=vi0(x,s), x∈R, s∈[−τi,0], i=1,2, $ | (2) |
where
$ \mathcal{D} [v_i](x,t) = v_i(x+1,t)-2v_i(x,t)+v_i(x-1,t),\ i = 1,2. $ |
Here
According to (H1), the spatially homogeneous system of (1) admits two constant equilibria
$ (v_{1-}, v_{2-}) = {\bf{0}}: = (0,0)\quad \mbox{and} \quad (v_{1+}, v_{2+}) = {\bf{K}}: = (K_1, K_2). $ |
It is clear that (H1) is a basic assumption to ensure that system (1) is monostable on
$ {cϕ′1(ξ)=d1D[ϕ1](ξ)−αϕ1(ξ)+h(ϕ2(ξ−cτ1)),cϕ′2(ξ)=d2D[ϕ2](ξ)−βϕ2(ξ)+g(ϕ1(ξ−cτ2)),(ϕ1,ϕ2)(−∞)=(v1−,v2−),(ϕ1,ϕ2)(+∞)=(v1+,v2+), $ | (3) |
where
System (1) is a discrete version of classical epidemic model
$ {∂tv1(x,t)=d1∂xxv1(x,t)−a1v1(x,t)+h(v2(x,t−τ1)),∂tv2(x,t)=d2∂xxv2(x,t)−a2v2(x,t)+g(v1(x,t−τ2)). $ | (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
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
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,t−τ)),∂tv2(x,t)=d2D[v2](x,t)−βv2(x,t)+g(v1(x,t−τ)),vi(x,s)=vi0(x,s), x∈R, 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(x−1,t)], x∈R, t>0,u(x,0)=u0(x), x∈R, $ |
can be expressed by
$ u(x,t)=(S(t)u0)(x)=e−2dt∞∑m=−∞Im(2dt)u0(x−m), $ |
where
$ Im(t)=∞∑k=0(t/2)m+2kk!(m+k)!, $ |
and
$ I′m(t)=12[Im+1(t)+Im−1(t)], ∀t>0,m∈Z, $ | (7) |
and
$ e−t∞∑m=−∞Im(t)=e−t[I0(t)+2I1(t)+2I2(t)+I3(t)+⋯]=1. $ | (8) |
Thus, the solution
$ {v1(x,t)=e−(2d1+α)t∑∞m=−∞Im(2d1t)v10(x−m,0)+∑∞m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))(h(v2(x−m,s−τ)))ds,v2(x,t)=e−(2d2+β)t∑∞m=−∞Im(2d2t)v20(x−m,0)+∑∞m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))(g(v1(x−m,s−τ)))ds. $ | (9) |
In fact, by [8,Lemma 2.1], we can differentiate the series on
$ ∂tv1(x,t)=−(2d1+α)e−(2d1+α)t∞∑m=−∞Im(2d1t)v10(x−m,0) +e−(2d1+α)t∞∑m=−∞2d1I′m(2d1t)v10(x−m,0) +∞∑m=−∞Im(0)(h(v2(x−m,t−τ))) −(2d1+α)∞∑m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))(h(v2(x−m,s−τ)))ds +∞∑m=−∞∫t0e−(2d1+α)(t−s)2d1I′m(2d1(t−s))(h(v2(x−m,s−τ)))ds=d1[v1(x+1,t)−2v1(x,t)+v1(x−1,t)]−αv1(x,t)+h(v2(x,t−τ)) $ |
and
$ ∂tv2(x,t)=−(2d2+β)e−(2d2+β)t∞∑m=−∞Im(2d2t)v20(x−m,0) +e−(2d2+β)t∞∑m=−∞2d2I′m(2d2t)v20(x−m,0) +∞∑m=−∞Im(0)(g(v1(x−m,t−τ))) −(2d2+β)∞∑m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))(g(v1(x−m,s−τ)))ds +∞∑m=−∞∫t0e−(2d2+β)(t−s)2d2I′m(2d2(t−s))(g(v1(x−m,s−τ)))ds=d2[v2(x+1,t)−2v2(x,t)+v2(x−1,t)]−βv2(x,t)+g(v1(x,t−τ)). $ |
Next we investigate the characteristic roots of the linearized system for the wave profile system (3) at the trivial equilibrium
$ P1(c,λ):=f1(c,λ)−f2(c,λ) $ |
for
$ f1(c,λ):=Δ1(c,λ)Δ2(c,λ),f2(c,λ):=h′(0)g′(0)e−2cλτ, $ |
with
$ Δ1(c,λ)=d1(eλ+e−λ−2)−cλ−α,Δ2(c,λ)=d2(eλ+e−λ−2)−cλ−β. $ |
It is easy to see that
Similar to [27,Lemma 3.1], we can obtain the following result.
Lemma 2.1. There exists a positive constant
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:
(i)
(ii)
(iii)
$ 0<g−(u)≤g(u)≤g+(u)≤g′(0)u for u∈[0,K+1],0<h−(v)≤h(v)≤h+(v)≤h′(0)v for v∈[0,K+2]. $ |
Proposition 1. Assume that
$ K−1≤lim infξ→+∞ϕ1(ξ)≤lim supξ→+∞ϕ1(ξ)≤K+1,0≤lim infξ→+∞ϕ2(ξ)≤lim supξ→+∞ϕ2(ξ)≤K+2. $ |
Finally, we shall state the stability result of traveling waves derived in Proposition 1. Before that, let us introduce the following notations.
Notations.
$ ||f||L1w(I)=∫Iw(x)|f(x)|dx, $ |
$ ||f||Wk,1w(I)=k∑i=0∫Iw(x)|dif(x)dxi|dx. $ |
Let
$ F[f](η)=ˆf(η)=∫Re−ixηf(x)dx $ |
and the inverse Fourier transform is given by
$ F−1[ˆf](x)=12π∫Reixηˆf(η)dη, $ |
where
To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.
$ limx→±∞(v10(x,s),v20(x,s))=(v1±,v2±) uniformly in s∈[−τ,0]. $ |
Consider the following function
$ \mathcal{P}_{2}(\lambda,c) = d_{2}(e^\lambda+e^{-\lambda}-2)-c\lambda-\beta +\max\{h'(0), g'(0)\}e^{- \lambda c\tau}. $ |
Since
We select the weight function
$ w(ξ)=e−2λξ, $ |
where
Theorem 2.2. (Global stability of traveling waves). Assume that
$ vi0(x,s)−ϕi(x+cs)∈Cunif[−τ,0]∩C([−τ,0];W1,1w(R)), i=1,2,∂s(vi0−ϕi)∈L1([−τ,0];L1w(R)), i=1,2, $ |
then there exists
$ supx∈R|vi(x,t)−ϕi(x+ct)|≤Ce−μt,t>0, $ |
where
$ Cunif[r,T]={u∈C([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
$ {Vi(ξ,t):=vi(x,t)−ϕi(x+ct)=vi(ξ−ct,t)−ϕi(ξ), i=1,2,Vi0(ξ,s):=vi0(x,s)−ϕi(x+cs)=vi0(ξ−cs,s)−ϕ(ξ), i=1,2. $ |
Then it follows from (1) and (3) that
$ {V1t+cV1ξ−d1D[V1]+αV1=Q1(V2(ξ−cτ,t−τ)),V2t+cV2ξ−d2D[V2]+βV2=Q2(V1(ξ−cτ,t−τ)),Vi(ξ,s)=Vi0(ξ,s), (ξ,s)∈R×[−τ,0], i=1,2. $ | (10) |
The nonlinear terms
$ {Q1(V2):=h(ϕ2+V2)−h(ϕ2)=h′(˜ϕ2)V2,Q2(V1):=g(ϕ1+V1)−g(ϕ1)=g′(˜ϕ1)V1, $ | (11) |
for some
We first prove the existence and uniqueness of solution
Lemma 3.1. Assume that
Proof. Let
$ {U1t−d1D[U1]+αU1=Q1(U2(x,t−τ)),U2t−d2D[U2]+βU2=Q2(U1(x,t−τ)),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
$ {U1(x,t)=e−(2d1+α)t∑∞m=−∞Im(2d1t)U10(x−m,0) +∑∞m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))Q1(U20(x−m,s−τ))ds,U2(x,t)=e−(2d2+β)t∑∞m=−∞Im(2d2t)U20(x−m,0) +∑∞m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))Q2(U10(x−m,s−τ))ds $ | (13) |
for
Since
$ limx→+∞U1(x,t)=e−(2d1+α)t∞∑m=−∞Im(2d1t)limx→+∞U10(x−m,0)+∞∑m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))limx→+∞Q1(U20(x−m,s−τ))ds=e−αtU10(∞,0)+∫t0e−α(t−s)Q1(U20(∞,s−τ))∞∑m=−∞e−2d1(t−s)Im(2d1(t−s))ds=:U1(t) uniformly in t∈[0,τ] $ | (14) |
and
$ limx→+∞U2(x,t)=e−(2d2+β)t∞∑m=−∞Im(2d2t)limx→+∞U20(x−m,0) $ |
$ +∞∑m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))limx→+∞Q2(U10(x−m,s−τ))ds=e−βtU20(∞,0)+∫t0e−β(t−s)Q2(U10(∞,s−τ))∞∑m=−∞e−2d2(t−s)Im(2d2(t−s))ds=:U2(t) uniformly in t∈[0,τ], $ | (15) |
where we have used (8). Thus, we obtain that
When
$ U1(x,t)=e−(2d1+α)(t−τ)∞∑m=−∞Im(2d1(t−τ))U1(x−m,τ)+∞∑m=−∞∫tτe−(2d1+α)(t−s)Im(2d1(t−s))Q1(U2(x−m,s−τ))ds,U2(x,t)=e−(2d2+β)(t−τ)∞∑m=−∞Im(2d2(t−τ))U2(x−m,τ)+∞∑m=−∞∫tτe−(2d2+β)(t−s)Im(2d2(t−s))Q2(U1(x−m,s−τ))ds. $ |
Similarly, by (14) and (15), we have
$ limx→+∞U1(x,t)=e−(2d1+α)(t−τ)∞∑m=−∞Im(2d1(t−τ))limx→+∞U1(x−m,τ)+∞∑m=−∞∫tτe−(2d1+α)(t−s)Im(2d1(t−s))limx→+∞Q1(U2(x−m,s−τ))ds=e−α(t−τ)U1(τ)+∫tτe−α(t−s)Q1(U1(s−τ))∞∑m=−∞e−2d1(t−s)Im(2d1(t−s))ds=:ˉU1(t) uniformly in t∈[τ,2τ], $ |
and
$ limx→+∞U2(x,t)=e−(2d2+β)(t−τ)∞∑m=−∞Im(2d2(t−τ))limx→+∞U2(x−m,τ)+∞∑m=−∞∫tτe−(2d2+β)(t−s)Im(2d2(t−s))limx→+∞Q2(U1(x−m,s−τ))ds=e−β(t−τ)U2(τ)+∫tτe−β(t−s)Q2(U2(s−τ))∞∑m=−∞e−2d2(t−s)Im(2d2(t−s))ds $ |
$ =:ˉU2(t) uniformly in t∈[τ,2τ]. $ |
By repeating this procedure for
Now we state the stability result for the perturbed system (10), which automatically implies Theorem 2.2.
Proposition 2. Assume that
$ Vi0∈Cunif[−τ,0]∩C([−τ,0];W1,1w(R)), i=1,2, $ |
and
$ ∂sVi0∈L1([−τ,0];L1w(R)), i=1,2, $ |
then there exists
$ supξ∈R|Vi(ξ,t)|≤Ce−μt,t>0, i=1,2, $ | (16) |
for some
In order to prove Proposition 2, we first investigate the decay estimate of
Lemma 3.2. Assume that
$ supξ∈[x0,+∞)|Vi(ξ,t)|≤Ce−μ1t, t>0, i=1,2, $ |
for some
Proof. Denote
$ z+i(t):=Vi(∞,t), z+i0(s):=Vi0(∞,s), s∈[−τ,0], i=1,2. $ |
Since
$ \lim\limits_{\xi\rightarrow+\infty}V_{i}(\xi,t) = z_{i}^{+}(t) $ |
exists uniformly for
$ {dz+1dt+αz+1−h′(v2+)z+2(t−τ)=P1(z+2(t−τ)),dz+2dt+βz+2−g′(v1+)z+1(t−τ)=P2(z+1(t−τ)),z+i(s)=z+i0(s), s∈[−τ,0], i=1,2, $ |
where
$ {P1(z+2)=h(v2++z+2)−h(v2+)−h′(v2+)z+2,P2(z+1)=g(v1++z+1)−g(v1+)−g′(v1+)z+1. $ |
Then by [9,Lemma 3.8], there exist positive constants
$ |Vi(∞,t)|=|z+i(t)|≤Ce−μ1t, t>0, i=1,2, $ | (17) |
provided that
By the continuity and the uniform convergence of
$ \sup\limits_{\xi\in[x_{0},+\infty)}|V_{i}(\xi,t)|\leq Ce^{-\mu_{1}t}, \ t > 0, \ i = 1,2, $ |
provided that
$ \lim\limits_{x\rightarrow +\infty}(v_{10}(x,s), v_{20}(x,s)) = (K_{1}, K_{2}) \ \mbox{uniformly in} \ s\in[-\tau,0], $ |
which implies
$ \lim\limits_{\xi\rightarrow +\infty}V_{i0}(\xi,s) = \lim\limits_{\xi\rightarrow +\infty}[v_{i0}(\xi,s)-\phi_{i}(\xi)] = K_{i}-K_{i} = 0 $ |
uniformly for
Next we are going to establish the a priori decay estimate of
$ ˜Vi(ξ,t)=√w(ξ)Vi(ξ+x0,t)=e−λξVi(ξ+x0,t),i=1,2. $ |
Substituting
$ {˜V1t+c˜V1ξ+c1˜V1(ξ,t)−d1eλ˜V1(ξ+1,t)−d1e−λ˜V1(ξ−1,t)=˜Q1(˜V2(ξ−cτ,t−τ)),˜V2t+c˜V2ξ+c2˜V2(ξ,t)−d2eλ˜V2(ξ+1,t)−d2e−λ˜V2(ξ−1,t)=˜Q2(˜V1(ξ−cτ,t−τ)),˜Vi(ξ,s)=√w(ξ)Vi0(ξ+x0,s)=:˜Vi0(ξ,s), ξ∈R,s∈[−τ,0], i=1,2, $ | (18) |
where
$ c1=cλ+2d1+α,c2=cλ+2d2+β $ |
and
$ ˜Q1(˜V2)=e−λξQ1(V2),˜Q2(˜V1)=e−λξQ2(V1). $ |
By (11),
$ ˜Q1(˜V2(ξ−cτ,t−τ))=e−λξQ1(V2(ξ−cτ+x0,t−τ))=e−λξh′(˜ϕ2)V2(ξ−cτ+x0,t−τ)=e−λcτh′(˜ϕ2)˜V2(ξ−cτ,t−τ) $ | (19) |
and
$ ˜Q2(˜V1(ξ−cτ,t−τ))=e−λcτg′(˜ϕ1)˜V1(ξ−cτ,t−τ). $ | (20) |
By (H3), we further obtain
$ |˜Q1(˜V2(ξ−cτ,t−τ))|≤h′(0)e−λcτ|˜V2(ξ−cτ,t−τ)|,|˜Q2(˜V1(ξ−cτ,t−τ))|≤g′(0)e−λcτ|˜V1(ξ−cτ,t−τ)|. $ |
Taking (19) and (20) into (18), one can see that the coefficients
$ {V+1t+cV+1ξ+c1V+1(ξ,t)−d1eλV+1(ξ+1,t)−d1e−λV+1(ξ−1,t) =h′(0)e−λcτV+2(ξ−cτ,t−τ),V+2t+cV+2ξ+c2V+2(ξ,t)−d2eλV+2(ξ+1,t)−d2e−λV+2(ξ−1,t) =g′(0)e−λcτV+1(ξ−cτ,t−τ), $ | (21) |
with
$ V^+_i(\xi,s) = \sqrt{w(\xi)}V_{i0}(\xi+x_0,s) = :V^+_{i0}(\xi,s), \ i = 1,2, $ |
where
Now we are in a position to derive the decay estimate of
Lemma 3.3. ([11,Lemma 3.1]) Let
$ {ddtz(t)=Az(t)+Bz(t−τ),t≥0,τ>0,z(s)=z0(s),s∈[−τ,0]. $ | (22) |
where
$ z(t)=eA(t+τ)eB1tτz0(−τ)+∫0−τeA(t−s)eB1(t−τ−s)τ[z′0(s)−Az0(s)]ds, $ |
where
$ eB1tτ={0,−∞<t<−τ,I,−τ≤t<0,I+B1t1!,0≤t<τ,I+B1t1!+B21(t−τ)22!,τ≤t<2τ,⋮⋮I+B1t1!+B21(t−τ)22!+⋯+Bm1[t−(m−1)τ]mm!,(m−1)τ≤t<mτ,⋮⋮ $ |
where
Lemma 3.4. ([11,Theorem 3.1]) Suppose
$ ‖z(t)‖≤C0e−ετσt,t>0, $ |
where
$ ‖eAteB1tτ‖≤C0e−ετσt,t>0, $ |
where
From the proof of [11,Theome 3.1], one can see that
$ μ1(A)=limθ→0+‖I+θA‖−1θ=max1≤j≤N[Re(ajj)+N∑j≠i|aij|] $ |
and
$ μ∞(A)=limθ→0+‖I+θA‖∞−1θ=max1≤i≤N[Re(aii)+N∑i≠j|aij|]. $ |
Taking the Fourier transform to (21) and denoting the Fourier transform of
$ {∂∂tˆV+1(η,t)=(−c1+d1(eλ+iη+e−(λ+iη))−icη)ˆV+1(η,t) +h′(0)e−cτ(λ+iη)ˆV+2(η,t−τ),∂∂tˆV+2(η,t)=(−c2+d2(eλ+iη+e−(λ+iη))−icη)ˆV+2(η,t) +g′(0)e−cτ(λ+iη)ˆV+1(η,t−τ),ˆV+i(η,s)=ˆV+i0(η,s), η∈R, s∈[−τ,0], i=1,2. $ | (23) |
Let
$ A(\eta) = \left( −c1+d1(eλ+iη+e−(λ+iη))−icη00−c2+d2(eλ+iη+e−(λ+iη))−icη \right) $ |
and
$ B(\eta) = \left( 0h′(0)e−cτ(λ+iη)g′(0)e−cτ(λ+iη)0 \right). $ |
Then system (23) can be rewritten as
$ ˆV+t(η,t)=A(η)ˆV+(η,t)+B(η)ˆV+(η,t−τ). $ | (24) |
By Lemma 3.3, the linear delayed system (24) can be solved by
$ ˆV+(η,t)=eA(η)(t+τ)eB1(η)tτˆV+0(η,−τ)+∫0−τeA(η)(t−s)eB1(η)(t−s−τ)τ[∂sˆV+0(η,s)−A(η)ˆV+0(η,s)]ds:=I1(η,t)+∫0−τI2(η,t−s)ds, $ | (25) |
where
$ V+(ξ,t) $ | (26) |
$ =F−1[I1](ξ,t)+∫0−τF−1[I2](ξ,t−s)ds=12π∫∞−∞eiξηeA(η)(t+τ)eB1(η)tτˆV+0(η,−τ)dη +12π∫0−τ∫∞−∞eiξηeA(η)(t−s)eB1(η)(t−s−τ)τ[∂sˆV+0(η,s)−A(η)ˆV+0(η,s)]dηds. $ | (27) |
Lemma 3.5. Let the initial data
$ V_{i0}^{+}\in C([-\tau,0];W^{1,1}(\mathbb{R})),\ \partial_{s}V_{i0}^{+}\in L^1([-\tau, 0]; L^{1}(\mathbb{R})),\ i = 1,2. $ |
Then
$ ‖V+i(t)‖L∞(R)≤Ce−μ2t for c≥max{c∗,c∗}, i=1,2, $ |
where
Proof. According to (26), we shall estimate
$ μ(A(η))=μ1(A(η))+μ∞(A(η))2=max{−c1+d1(eλcosη+e−λcosη),−c2+d2(eλcosη+e−λcosη)}=−c2+d2(eλcosη+e−λcosη)=−c2+d2(eλ+e−λ)cosη=−cλ+d2(eλ+e−λ−2)−β−m(η), $ |
where
$ m(\eta) = d_2(1-\cos\eta)(e^\lambda+e^{-\lambda})\ge0, $ |
since
$ ν(B(η))=max{h′(0),g′(0)}e−λcτ. $ |
By considering
$ \mu(A(\eta))+\nu(B(\eta)) = -c\lambda+d_2(e^\lambda+e^{-\lambda}-2)-\beta-m(\eta)+\max\{h'(0), g'(0)\}e^{- \lambda c\tau} < 0. $ |
Furthermore, we obtain
$ |μ(A(η))|−ν(B(η))=cλ−d2(eλ+e−λ−2)+β+m(η)−max{h′(0),g′(0)}e−λcτ=−P2(λ,c)+m(η), $ |
where
$ ‖eA(η)(t+τ)eB1(η)t‖≤C1e−ετ(|μ(A(η))|−ν(B(η)))t≤C1e−ετμ0te−ετm(η)t, $ | (28) |
where
$ \sup\limits_{\eta\in\mathbb{R}}\|\hat{V}_{0}^{+}(\eta,-\tau)\| \leq\int_{\mathbb{R}}\|V_{0}^{+}(\xi,-\tau)\|d\xi = \sum\limits_{i = 1}^2\|V_{i0}^{+}(\cdot, -\tau)\|_{L^{1}(\mathbb{R})}. $ |
Applying (28), we derive
$ supξ∈R‖F−1[I1](ξ,t)‖=supξ∈R‖12π∫∞−∞eiξηeA(η)(t+τ)eB1(η)tˆV+0(η,−τ)dη‖≤C∫∞−∞e−ετm(η)te−ετμ0t‖ˆV+0(η,−τ)‖dη≤Ce−ετμ0tsupη∈R‖ˆV+0(η,−τ)‖∫∞−∞e−ετm(η)tdη≤Ce−μ2t2∑i=1‖V+i0(⋅,−τ)‖L1(R), $ | (29) |
with
Note that
$ \sup\limits_{\eta\in\mathbb{R}}\|A(\eta)\hat{V}_{0}^{+}(\eta,s)\| \leq C\sum\limits_{i = 1}^{2}\|V_{i0}^{+}(\cdot, s)\|_{W^{1,1}(\mathbb{R})}. $ |
Similarly, we can obtain
$ supξ∈R‖F−1[I2](ξ,t−s)‖=supξ∈R‖12π∫∞−∞eiξηeA(η)(t−s)eB1(η)(t−s−τ)[∂sˆV+0(η,s)−A(η)ˆV+0(η,s)]dη‖≤C∫∞−∞e−ετm(η)(t−s)e−ετμ0(t−s)‖∂sˆV+0(η,s)−A(η)ˆV+0(η,s)‖dη≤Ce−ετμ0teετμ0ssupη∈R‖∂sˆV+0(η,s)−A(η)ˆV+0(η,s)‖∫∞−∞e−ετm(η)(t−s)dη. $ |
It then follows that
$ ∫0−τsupξ∈R‖F−1[I2](ξ,t−s)‖ds≤Ce−ετμ0t∫0−τeετμ0ssupη∈R‖∂sˆV+0(η,s)−A(η)ˆV+0(η,s)‖∫∞−∞e−ετm(η)(t−s)dηds≤Ce−ετμ0t∫0−τ‖∂sV+0(⋅,s)‖L1(R)+‖V+0(⋅,s)‖W1,1(R)ds≤Ce−ετμ0t(‖∂sV+0(s)‖L1([−τ,0];L1(R))+‖V+0(s)‖L1([−τ,0];W1,1(R))). $ | (30) |
Substituting (29) and (30) to (26), we obtain the following the decay rate
$ 2∑i=1‖V+i(t)‖L∞(R)≤Ce−μ2t. $ |
This proof is complete.
The following maximum principle is needed to obtain the crucial boundedness estimate of
Lemma 3.6. Let
$ {∂v∂t+a1∂v∂ξ+a2v−deνv(t,ξ+1)−de−νv(t,ξ−1)≥0, (t,ξ)∈(0,T]×R,v(0,ξ)≥0,ξ∈R, $ | (31) |
then
Lemma 3.7. When
Proof. When
$ h′(0)e−λcτV+2(ξ−cτ,t−τ)=h′(0)e−λcτV+20(ξ−cτ,t−τ)≥0. $ | (32) |
Applying (32) to the first equation of (21), we get
$ {V+1t+cV+1ξ+c1V+1(ξ,t)−d1eλV+1(ξ+1,t)−d1e−λV+1(ξ−1,t)≥0, (ξ,t)∈R×[0,τ],V+10(ξ,s)≥0, ξ∈R, s∈[−τ,0]. $ |
By Lemma 3.6, we derive
$ V+1(ξ,t)≥0,(ξ,t)∈R×[0,τ]. $ | (33) |
Similarly, we obtain
$ {V+2t+cV+2ξ+c2V+2(ξ,t)−d2eλV+2(ξ+1,t)−d2e−λV+2(ξ−1,t)≥0, (ξ,t)∈R×[0,τ],V+20(ξ,s)≥0, ξ∈R s∈[−τ,0]. $ |
Using Lemma 3.6 again, we obtain
$ V+2(ξ,t)≥0,(ξ,t)∈R×[0,τ]. $ | (34) |
When
$ (V+1(ξ,t),V+2(ξ,t))≥(0,0),(ξ,t)∈R×[nτ,(n+1)τ]. $ | (35) |
Combining (33), (34) and (31), we obtain
Now we establish the following crucial boundedness estimate for
Lemma 3.8. Let
$ |˜Vi0(ξ,s)|≤V+i0(ξ,s)for(ξ,s)∈R×[−τ,0], i=1,2, $ | (36) |
then
$ |˜Vi(ξ,t)|≤V+i(ξ,t)for(ξ,t)∈R×[0,+∞), i=1,2. $ |
Proof. First of all, we prove
$ |˜Vi(ξ−cτ,t−τ)|=|˜Vi0(ξ−cτ,t−τ)|≤V+i0(ξ−cτ,t−τ)=V+i(ξ−cτ,t−τ)for (ξ,t)∈R×[0,τ]. $ | (37) |
Then by
$ h′(0)e−λcτV+2(ξ−cτ,t−τ)±h′(˜ϕ2)e−λcτ˜V2(ξ−cτ,t−τ)≥h′(0)e−λcτV+2(ξ−cτ,t−τ)−|h′(˜ϕ2)|e−λcτ|˜V2(ξ−cτ,t−τ)|≥0for (ξ,t)∈R×[0,τ] $ | (38) |
and
$ g′(0)e−λcτV+1(ξ−cτ,t−τ)±g′(˜ϕ1)e−λcτ˜V1(ξ−cτ,t−τ)≥0for (ξ,t)∈R×[0,τ]. $ | (39) |
Let
$ U_{i}^{-}(\xi,t): = V_{i}^{+}(\xi,t)-\widetilde{V}_{i}(\xi,t)\quad and \quad U_{i}^{+}(\xi,t): = V_{i}^{+}(\xi,t)+\widetilde{V}_{i}(\xi,t),\quad i = 1,2. $ |
We are going to estimate
From (18), (19), (21) and (38), we see that
$ {U−1t+cU−1ξ+c1U−1(ξ,t)−d1eλU−1(ξ+1,t)−d1e−λU−1(ξ−1,t)≥0,(ξ,t)∈R×[0,τ],U−10(ξ,s)=V+10(ξ,s)−˜V10(ξ,s)≥0,ξ∈R, s∈[−τ,0]. $ |
By Lemma 3.6, we obtain
$ U^{-}_{1}(\xi,t) \geq 0, \quad (\xi,t)\in \mathbb{R}\times[0,\tau], $ |
namely,
$ ˜V1(ξ,t)≤V+1(ξ,t),(ξ,t)∈R×[0,τ]. $ | (40) |
Similarly, one has
$ {U−2t+cU−2ξ+c2U−2(ξ,t)−d2eλU−2(ξ+1,t)−d2e−λU−2(ξ−1,t)≥0,(ξ,t)∈R×[0,τ],U−20(ξ,s)=V+20(ξ,s)−˜V20(ξ,s)≥0,ξ∈R, s∈[−τ,0]. $ |
Applying Lemma 3.6 again, we have
$ U^{-}_{2}(\xi,t)\geq 0,\quad (\xi,t)\in\mathbb{R}\times[0,\tau], $ |
i.e.,
$ ˜V2(ξ,t)≤V+2(ξ,t),(ξ,t)∈R×[0,τ]. $ | (41) |
On the other hand,
$ {U+1t+cU+1ξ+c1U+1(ξ,t)−d1eλU+1(ξ+1,t)−d1e−λU+1(ξ−1,t)≥0,(ξ,t)∈R×[0,τ],U−10(ξ,s)=V+10(ξ,s)−˜V10(ξ,s)≥0,ξ∈R, s∈[−τ,0]. $ |
Then Lemma 3.6 implies that
$ U^{+}_{1}(\xi,t) = V^{+}_{1}(\xi,t)+\widetilde{V}_{1}(\xi,t)\geq 0,\quad (\xi,t)\in \mathbb{R}\times[0,\tau], $ |
that is,
$ −V+1(ξ,t)≤˜V1(ξ,t),(ξ,t)∈R×[0,τ]. $ | (42) |
Similarly,
$ {U+2t+cU+2ξ+c2U+2(ξ,t)−d2eλU+2(ξ+1,t)−d2e−λU+2(ξ−1,t)≥0,(ξ,t)∈R×[0,τ],U−20(ξ,s)=V+20(ξ,s)−˜V10(ξ,s)≥0,ξ∈R, s∈[−τ,0]. $ |
Therefore, we can prove that
$ U^{+}_{2}(\xi,t) = V^{+}_{2}(\xi,t)+\widetilde{V}_{2}(\xi,t)\geq 0,\quad (\xi,t)\in \mathbb{R}\times[0,\tau], $ |
namely
$ −V+2(ξ,t)≤˜V2(ξ,t),(ξ,t)∈R×[0,τ]. $ | (43) |
Combining (40) and (42), we obtain
$ |˜V1(ξ,t)|≤V+1(ξ,t)for(ξ,t)∈R×[0,τ], $ | (44) |
and combining (41) and (43), we prove
$ |˜V2(ξ,t)|≤V+2(ξ,t)for(ξ,t)∈R×[0,τ]. $ | (45) |
Next, when
$ |˜Vi(ξ,t)|≤V+i(ξ,t)for(ξ,t)∈R×[τ,2τ],i=1,2. $ |
Repeating this procedure, we then further prove
$ |˜Vi(ξ,t)|≤V+i(ξ,t),(ξ,t)∈R×[nτ,(n+1)τ],n=1,2,⋯, $ |
which implies
$ |˜Vi(ξ,t)|≤V+i(ξ,t)for(ξ,t)∈R×[0,∞),i=1,2. $ |
The proof is complete.
Let us choose
$ V_{i0}^{+}\in C([-\tau,0];W^{1,1}(\mathbb{R})), \quad \partial_{s}V_{i0}^{+}\in L^1([-\tau,0]; L^{1}(\mathbb{R})), $ |
and
$ V_{i0}^{+}(\xi,s)\geq |V_{i0}(\xi,s)|,\quad (\xi,s)\in \mathbb{R}\times[-\tau,0],\ i = 1,2. $ |
Combining Lemmas 3.5 and 3.8, we can get the convergence rates for
Lemma 3.9. When
$ ‖˜Vi(t)‖L∞(R)≤Ce−μ2t, $ |
for some
Lemma 3.10. It holds that
$ supξ∈(−∞,x0]|Vi(ξ,t)|≤Ce−μ2t, i=1,2, $ |
for some
Proof. Since
$ supξ∈(−∞,0]|Vi(ξ+x0,t)|≤‖˜Vi(t)‖L∞(R)≤Ce−μ2t, $ |
which implies
$ supξ∈(−∞,x0]|Vi(ξ,t)|≤Ce−μ2t. $ |
Thus, the estimate for the unshifted
Proof of Proposition 3.2. By Lemmas 3.2 and 3.10, we immediately obtain (16) for
We are grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.
[1] | Acierno R, Hernandez MA, Amstadter AB, et al. (2010) Prevalence and correlates of emotional, physical, sexual, and financial abuse and potential neglect in the United States: The National Elder Mistreatment Study. Am J Public Health 100(2): 292. |
[2] | Dong X, Simon MA, Evans DA. (2014) A population-based study of physical function and risk for elder abuse reported to social services agency: Findings from the Chicago Health and Aging Project. J Applied Gerontol 33(7): 808-830. |
[3] | Dong XQ, Simon MA, Rajan K, et al. (2012) Association of cognitive function and risk for elder abuse in a community-dwelling population. Dement Geriatr Cogn Disord 32(3): 209-215. |
[4] | Dong X, Simon M, Evans D. (2012) Decline in physical function and risk of elder abuse reported to social services in a community-dwelling population of older adults. J Am Geriatr Society60(10): 1922-1928. |
[5] | Dong X, Gorbien MJ, Simon MA, et al. (2006) Loneliness and risk of elder mistreatment in older people in China. J Am Geriatr Society 54(4s): 27-28. |
[6] | Dong X, Simon MA, Odwazny R, et al. (2008) Depression and elder abuse and neglect among community-dwelling Chinese elderly population. J Elder Abuse Neglect 20(1): 25-41. |
[7] | Dong X, Chang ES, Wong E, et al. (2014) Association of depressive symptomatology and elder mistreatment in a US Chinese population: findings from a community-based participatory research study. J Agress Maltreat Trauma 23(1): 81-98. |
[8] | Dong X, Chen R, Chang ES, et al. (2012) Elder abuse and psychological well-being: A systematic review and implications for research and policy-A mini review. Gerontology 59(2):132-142. |
[9] | Dong X, Simon M, Mendes de Leon C, et al. (2009) Elder self-neglect and abuse and mortality risk in a community-dwelling population. JAMA 302(5): 517-526. |
[10] | Dong X, Simon MA. (2013) Elder abuse as a risk factor for hospitalization in older persons. JAMA Internal Med 173(10): 911-917. |
[11] | Dong X, Simon MA, Evans D. (2012) Prospective study of the elder self-neglect and ED use in a community population. Am J Emerg Med 30(4): 553-561. |
[12] | Dong X, Simon MA. (2013) Association between reported elder abuse and rates of admission to skilled nursing facilities: findings from a longitudinal population-based cohort study. Gerontology 59(5): 464-472. |
[13] | Garrer-Olmo J, Planas-Pujol X, Lopez-Pousa S, et al. (2009) Prevalence and risk factors of suspected elder abuse subtypes in people aged 75 and older. J Am Geriatr Society 57(5): 815-822. |
[14] | Amstadter AB, Zajac K, Strachan M, et al. (2011) Prevalence and correlates of elder mistreatment in south Carolina: the south Carolina elder mistreatment study. J Interpers Violence26(15): 2947-2972. |
[15] | Beach SR, Schulz R, Castle NG, et al. (2010) Financial exploitation and psychological mistreatment among older adults: differences between African Americans and Non-African Americans in a population-based survey. Gerontologist 50(6): 744-757. |
[16] | Pillemer K, Finkelhor D. (1988) The prevalence of elder abuse: a random sample survey. Gerontologist 28(1): 51-57. |
[17] | Dong X. (2014) Do the definitions of elder mistreatment subtypes matter? Findings from the PINE study. J Gerontol: Med Sci 69(2): 68-75. |
[18] | Dong X, Chang E-S, Wong E, et al. (2011) How do US Chinese older adults view elder mistreatment? Findings from a community-based participatory research study. J Aging Health23(2): 289-312. |
[19] | Moon A, Tomita SK, Jung-Kamei S. (2002) Elder mistreatment among four Asian American groups: An exploratory study on tolerance, victim blaming and attitudes toward third-party intervention. J Gerontological Social Work 36(1-2): 153-169. |
[20] | Dong X, Chang E-S, Wong E, et al. (2010) How do US Chinese older adults view elder mistreatment? Findings from a community-based participatory research study. J Aging Health0898264310385931. |
[21] | Dong X, Chen R, Fulmer T, et al. (2014) Prevalence and correlates of elder mistreatment in a community-dwelling population of US Chinese older adults. J Aging Health 2014;26(7):1209-1224. |
[22] | United States Census Bureau. (2010) American Fact Finder. Available from: http://factfinder2. census. gov/faces/nav/jsf/pages/index. xhtml. |
[23] | Mui AC, Kang SY, Kang D, et al. (2007) English language proficiency and health-related quality of life among Chinese and Korean immigrant elders. Health Social Work 32(2): 119-127. |
[24] | Simon M, Li Y, Dong X. (2014) Levels of health literacy in a community-dwelling population of Chinese older adults. J Gerontol: Med Sci 69(2): 54-60. |
[25] | Dong X, Li Y, Simon M. (2014) Social engagement among U. S. Chinese older adults-findings from the PINE study. The J Gerontology: Med Sci 69(2): 82-89. |
[26] | Dong X, Zhang M, Simon M. (2014) The prevalence of Cardiopulmony symptoms among Chinese older adults in the greater Chicago area. J Gerontol: Med Sci 2014; 69(2): 39-45. |
[27] | Simon M, Chang E, Zhang M, et al. (2014) The prevalence of loneliness among U. S. Chinese older adults. J Aging Health 26(7): 1172-1188. |
[28] | Dong X, Li Y, Simon M. (2014) Preventive care service usage among Chinese older adults in the greater Chicago area. J Gerontol A: Biol Sci: Med Sci 69(2): 68-75. |
[29] | Dong X, Chen R, Simon M. (2014) Experiences of discrimination among U. S. Chinese older adults. J Gerontol Med Sci 69(2): 76-81. |
[30] | Chen R, Simon M, Xinqi D. (2014) Gender difference in depressive symptoms in U. S. Chinese older adults. AIMS Med Sci 1(1):13-27. |
[31] | Dong X, Chen R, Wong E, et al. (2014) Suicidal ideation in an older US Chinese population. J Aging Health 26(7): 1189-1208. |
[32] | Dong X, Chen R, Chang ES, et al. (2014) The prevalence of suicide attempts among community-dwelling US Chinese older adults confindings from the PINE study. Ethn Inequal Health Social Care 7(1): 23-35. |
[33] | Simon M, Chen R, Chang E, et al. (2014) The association bettern filial piety and suicidal ideation: Findings from a community-dwelling Chinese aging population. J Gerontol: Med Sci 69(2):90-97. |
[34] | Dong X, Chen R, Simon M. (2014) Anxiety among community-dwelling U. S. Chinese older adults. J Gerontol: Med Sci 69(2): 61-67. |
[35] | Dong X, Chen R, Li C, et al. (2011) Understanding depressive symptoms among community-dwelling Chinese older adults in the greater Chicago area. J Aging Health26(7):1155-1171. |
[36] | Dong X, Chang E-S, Wong E, et al. (2014) Association of depressive symptomatology and elder mistreatment in a US Chinese population: findings from a community-based participatory research study. J Aggres Maltreat Trauma 23(1): 81-98. |
[37] | Dong X. (2014) The population study of Chinese elderly in Chicago. J Aging Health 26(7):1079-1084. |
[38] |
Dong X, Chang E-S, Simon M, et al. (2011) Sustaining community-university partnerships: lessons learned from a participatory research project with elderly Chinese. Gateways: Int J Commun Res Engag 4: 31-47. doi: 10.5130/ijcre.v4i0.1767
![]() |
[39] | Dong X, Chang E-S, Simon M, et al. (2014) Addressing health and well-being of U. S. Chinese older adults through community-based participatory research: Introduction to the PINE study. J Gerontol A Biol Sci Med Sci 69(Suppl 2): S1-S6. |
[40] | Dong X, Chang E-S, Wong E, et al. (2014) Working with culture: lessons learned from a community-engaged project in a Chinese aging population. Aging Health 7(4): 529-537. |
[41] | Dong X, Chang E-S, Wong E, et al. (2011) Working with culture: lessons learned from a community-engaged project in a Chinese aging population. Aging Health 7(4): 529-537. |
[42] | Dong X, Wong E, Simon MA. (2014) Study design and implementation of the PINE study. J Aging Health 26(7): 1085-1099. |
[43] | Simon M, Chang E-S, Rajan KB, et al. (2014) Demographic characteris of U. S. Chinese older adults in the greater Chicago area: assessing the representativeness of the PINE study. J Aging Health 26(7): 1100-1115. |
[44] | Straus MA, Hamby SL, Boney-McCoy S, et al. (1996) The revised conflict tactics scales (CTS2) development and preliminary psychometric data. J Family Issues 17(3): 283-316. |
[45] | Katz S. (1983) Assessing self-maintenance: activities of daily living, mobility, and instrumental activities of daily living. J Am Geriatr Society |
[46] | Conrad KJ, Iris M, Ridings JW, et al. (2008) Conceptualizing and measuring financial exploitation of older adults. Report to the National Institute of Justice. 2006-MU-2004. |
[47] | Lachs MS, Williams C, O'Brien S, et al. (1997) Risk factors for reported elder abuse and neglect: a nine-year observational cohort study. Gerontologist 37(4): 469-474. |
[48] | Pillemer K, Finkelhor D. (1988) The prevalence of elder abuse: A random sample survey. Gerontologist 28(1): 51-57. |
[49] | Wu L, Chen H, Hu Y, et al. (2012) Prevalence and associated factors of elder mistreatment in a rural community in People's Republic of China: a cross-sectional study. PloS One 7(3): e33857. |
[50] | Breslau N, Davis GC, Andreski P, et al. (1991) Traumatic events and posttraumatic stress disorder in an urban population of young adults. Arch General Psychiatr 48(3): 216-222. |
[51] | Smith PH, Thornton GE, DeVellis R, et al. (2002) A population-based study of the prevalence and distinctiveness of battering, physical assault, and sexual assault in intimate relationships. Violence against Women 8(10): 1208-1232. |
[52] | Okazaki S. (2002) Influences of culture on Asian Americans' sexuality. J Sex Res 39(1): 34-41. |
[53] |
DeLiema M, Gassoumis ZD, Homeier DC, et al. (2012) Determining prevalence and correlates of elder abuse using promotores: low-income immigrant Latinos report high rates of abuse and neglect. J Am Geriatr Soc 60: 1333-1339. doi: 10.1111/j.1532-5415.2012.04025.x
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