The near vector space in which the additive inverse element does not necessarily exist is introduced in this paper. The reason is that an element in a near vector space which subtracts itself may not be a zero element. Therefore, the concept of a null set is introduced in this paper to play the role of a zero element. A near vector space can also be endowed with a norm to define a so-called near normed space. Based on this norm, the concept of a Cauchy sequence can be similarly defined. A near Banach space can also be defined according to the concept of completeness using the Cauchy sequences. The main aim of this paper is to establish the so-called near fixed point theorems and Meir-Keeler type of near fixed point theorems in near Banach spaces.
Citation: Hsien-Chung Wu. Near fixed point theorems in near Banach spaces[J]. AIMS Mathematics, 2023, 8(1): 1269-1303. doi: 10.3934/math.2023064
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The near vector space in which the additive inverse element does not necessarily exist is introduced in this paper. The reason is that an element in a near vector space which subtracts itself may not be a zero element. Therefore, the concept of a null set is introduced in this paper to play the role of a zero element. A near vector space can also be endowed with a norm to define a so-called near normed space. Based on this norm, the concept of a Cauchy sequence can be similarly defined. A near Banach space can also be defined according to the concept of completeness using the Cauchy sequences. The main aim of this paper is to establish the so-called near fixed point theorems and Meir-Keeler type of near fixed point theorems in near Banach spaces.
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
D[vi](x,t)=vi(x+1,t)−2vi(x,t)+vi(x−1,t), i=1,2. |
Here
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
{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
\begin{align*} \mathcal{F}[f](\eta) = \widehat f(\eta) = \int_\mathbb{R} e^{-\mathit{\boldsymbol{i}}x \eta}f(x)dx \end{align*} |
and the inverse Fourier transform is given by
\begin{align*} \mathcal{F}^{-1}[\widehat f](x) = \frac{1}{2\pi}\int_\mathbb{R}e^{\mathit{\boldsymbol{i}}x\eta }\widehat f(\eta)d\eta, \end{align*} |
where
To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.
\begin{align*} \lim\limits_{x\rightarrow \pm\infty}(v_{10}(x,s), v_{20}(x,s)) = (v_{1\pm}, v_{2\pm})\ \mbox{uniformly in }\ s\in[-\tau,0]. \end{align*} |
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
\begin{eqnarray*} w(\xi) = e^{-2\lambda \xi}, \end{eqnarray*} |
where
Theorem 2.2. (Global stability of traveling waves). Assume that
\begin{gather*} v_{i0}(x, s)-\phi_i(x + cs) \in C_{unif}[-\tau,0]\cap C([-\tau, 0]; W_w^{1,1}(\mathbb{R})),\ i = 1,2,\\ \partial_s(v_{i0}-\phi_i)\in L^1([-\tau,0]; L^1_w(\mathbb{R})),\ i = 1,2, \end{gather*} |
then there exists
\begin{align*} \sup\limits_{x\in\mathbb{R}} |v_i(x,t)-\phi_i(x + ct)|\le C e^{-\mu t}, \quad t > 0, \end{align*} |
where
\begin{align*} &C_{unif}[r,T]\\ = &\bigl\{{u\in C([r,T]\times \mathbb{R})} \mathit{\text{such that}} \big.\notag \big.\underset{x\rightarrow+\infty}{\lim} v(x,t) \mathit{\text{exists uniformly in}} t\in[r,T]\bigr\}. \end{align*} |
This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let
\begin{align*} \begin{cases} V_i(\xi, t): = v_i(x, t)- \phi_i(x+ct) = v_i(\xi-ct,t)-\phi_i(\xi),\ i = 1,2, \\ V_{i0}(\xi, s): = v_{i0}(x, s)- \phi_i(x+cs) = v_{i0}(\xi-cs,s)-\phi(\xi), \ i = 1,2. \end{cases} \end{align*} |
Then it follows from (1) and (3) that
\begin{align} \begin{cases} V_{1t}+c V_{1\xi}-d_1\mathcal{D}[V_{1}]+\alpha V_1 = Q_1(V_2(\xi-c\tau,t-\tau)),\\ V_{2 t}+cV_{2\xi}-d_2\mathcal{D}[V_{2}]+\beta V_2 = Q_2(V_1(\xi-c\tau,t-\tau)),\\ V_i(\xi,s) = V_{i0}(\xi,s), \ (\xi,s)\in\mathbb{R}\times[-\tau, 0],\ i = 1,2. \end{cases} \end{align} | (10) |
The nonlinear terms
\begin{align} \begin{cases} Q_1(V_2): = h(\phi_2 +V_2 )- h(\phi_2) = h'(\tilde \phi_2)V_2,\\ Q_2(V_1): = g(\phi_1 +V_1 )- g(\phi_1) = g'(\tilde \phi_1)V_1, \end{cases} \end{align} | (11) |
for some
We first prove the existence and uniqueness of solution
Lemma 3.1. Assume that
Proof. Let
\begin{align} \begin{cases} U_{1 t}-d_1\mathcal{D}[U_{1}]+\alpha U_1 = Q_1(U_2(x,t-\tau)),\\ U_{2 t}-d_2\mathcal{D}[U_{2}]+\beta U_2 = Q_2(U_1(x,t-\tau)),\\ U_i(x,s) = v_{i0}(x,s)-\phi_{i}(x+cs): = U_{i0}(x,s), \ (x,s)\in\mathbb{R}\times[-\tau, 0],\ i = 1,2. \end{cases} \end{align} | (12) |
Thus, the global existence and uniqueness of solution of (10) are transformed into that of (12).
When
\begin{align} \begin{cases} U_{1}(x,t) = e^{-(2d_1+\alpha)t}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_1 t)U_{10}(x-m,0) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum^{\infty}_{m = -\infty}\int_{0}^{t}e^{-(2d_1+\alpha)(t-s)}\mathbf{I}_{m}(2d_1(t-s))Q_1(U_{20}(x-m,s-\tau))ds,\\ U_{2}(x,t) = e^{-(2d_2+\beta)t}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_2 t)U_{20}(x-m,0) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum^{\infty}_{m = -\infty}\int_{0}^{t}e^{-(2d_2+\beta)(t-s)}\mathbf{I}_{m}(2d_2(t-s))Q_2(U_{10}(x-m,s-\tau))ds \end{cases} \end{align} | (13) |
for
Since
\begin{align} \begin{aligned} &\lim\limits_{x\rightarrow+\infty}U_{1}(x,t)\\ = &e^{-(2d_1+\alpha)t}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_1 t)\lim\limits_{x\rightarrow+\infty}U_{10}(x-m,0) \\ &+\sum^{\infty}_{m = -\infty}\int_{0}^{t}e^{-(2d_1+\alpha)(t-s)}\mathbf{I}_{m}(2d_1(t-s))\lim\limits_{x\rightarrow+\infty}Q_1(U_{20}(x-m,s-\tau))ds \\ = &e^{-\alpha t}U_{10}(\infty,0)+\int_{0}^{t}e^{-\alpha(t-s)}Q_1(U_{20}(\infty,s-\tau))\sum^{\infty}_{m = -\infty}e^{-2d_1(t-s)}\mathbf{I}_{m}(2d_1(t-s))ds \\ = &:\mathcal{U}_{1}(t) \ \ \mbox{uniformly in}\ t\in[0,\tau] \end{aligned} \end{align} | (14) |
and
\begin{align*} \begin{aligned} &\lim\limits_{x\rightarrow+\infty}U_{2}(x,t)\\ = &e^{-(2d_2+\beta)t}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_2t)\lim\limits_{x\rightarrow+\infty}U_{20}(x-m,0) \end{aligned} \end{align*} |
\begin{align} \begin{aligned}&+\sum^{\infty}_{m = -\infty}\int_{0}^{t}e^{-(2d_2+\beta)(t-s)}\mathbf{I}_{m}(2d_2(t-s))\lim\limits_{x\rightarrow+\infty}Q_2(U_{10}(x-m,s-\tau))ds \\ = &e^{-\beta t}U_{20}(\infty,0)+\int_{0}^{t}e^{-\beta(t-s)}Q_2(U_{10}(\infty,s-\tau))\sum^{\infty}_{m = -\infty}e^{-2d_2(t-s)}\mathbf{I}_{m}(2d_2(t-s))ds \\ = &:\mathcal{U}_{2}(t) \ \ \mbox{uniformly in}\ t\in[0,\tau], \end{aligned} \end{align} | (15) |
where we have used (8). Thus, we obtain that
When
\begin{align*} \begin{aligned} U_{1}(x,t) = &e^{-(2d_1+\alpha)(t-\tau)}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_1(t-\tau))U_{1}(x-m,\tau) \\ &+\sum^{\infty}_{m = -\infty}\int_{\tau}^{t}e^{-(2d_1+\alpha)(t-s)}\mathbf{I}_{m}(2d_1(t-s))Q_1(U_{2}(x-m,s-\tau))ds,\\ U_{2}(x,t) = &e^{-(2d_2+\beta)(t-\tau)}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_2(t-\tau))U_{2}(x-m,\tau) \\ &+\sum^{\infty}_{m = -\infty}\int_{\tau}^{t}e^{-(2d_2+\beta)(t-s)}\mathbf{I}_{m}(2d_2(t-s))Q_2(U_{1}(x-m,s-\tau))ds. \end{aligned} \end{align*} |
Similarly, by (14) and (15), we have
\begin{align*} \begin{aligned} &\lim\limits_{x\rightarrow+\infty}U_{1}(x,t)\\ = &e^{-(2d_1+\alpha)(t-\tau)}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_1(t-\tau))\lim\limits_{x\rightarrow+\infty}U_{1}(x-m,\tau) \\ &+\sum^{\infty}_{m = -\infty}\int_{\tau}^{t}e^{-(2d_1+\alpha)(t-s)}\mathbf{I}_{m}(2d_1(t-s))\lim\limits_{x\rightarrow+\infty}Q_1(U_{2}(x-m,s-\tau))ds\\ = &e^{-\alpha(t-\tau)}\mathcal{U}_1(\tau)+\int_{\tau}^{t}e^{-\alpha(t-s)}Q_1(\mathcal{U}_1(s-\tau)) \sum^{\infty}_{m = -\infty}e^{-2d_1(t-s)}\mathbf{I}_{m}(2d_1(t-s))ds \\ = &:\mathcal{\bar U}_{1}(t) \ \ \mbox{uniformly in}\ t\in[\tau,2\tau], \end{aligned} \end{align*} |
and
\begin{align*} \begin{aligned} &\lim\limits_{x\rightarrow+\infty}U_{2}(x,t)\\ = &e^{-(2d_2+\beta)(t-\tau)}\sum^{\infty}_{m = -\infty}\mathbf{I}_{m}(2d_2(t-\tau))\lim\limits_{x\rightarrow+\infty}U_{2}(x-m,\tau) \\ &+\sum^{\infty}_{m = -\infty}\int_{\tau}^{t}e^{-(2d_2+\beta)(t-s)}\mathbf{I}_{m}(2d_2(t-s))\lim\limits_{x\rightarrow+\infty}Q_2(U_{1}(x-m,s-\tau))ds\\ = &e^{-\beta(t-\tau)}\mathcal{U}_2(\tau)+\int_{\tau}^{t}e^{-\beta(t-s)}Q_2(\mathcal{U}_2(s-\tau)) \sum^{\infty}_{m = -\infty}e^{-2d_2(t-s)}\mathbf{I}_{m}(2d_2(t-s))ds \end{aligned} \end{align*} |
\begin{align*} \begin{aligned} = &:\mathcal{\bar U}_{2}(t) \ \ \mbox{uniformly in}\ t\in[\tau,2\tau]. \end{aligned} \end{align*} |
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
\begin{align*} V_{i0}\in C_{unif}[-\tau,0]\cap C([-\tau, 0]; W_w^{1,1}(\mathbb{R})), \ i = 1,2, \end{align*} |
and
\begin{align*} \partial_sV_{i0}\in L^1([-\tau, 0];L^1_w(\mathbb{R})),\ i = 1,2, \end{align*} |
then there exists
\begin{align} \sup\limits_{\xi\in\mathbb{R}} |V_i(\xi, t)|\le C e^{-\mu t}, \quad t > 0, \ i = 1,2, \end{align} | (16) |
for some
In order to prove Proposition 2, we first investigate the decay estimate of
Lemma 3.2. Assume that
\begin{align*} \sup\limits_{\xi\in [x_0,+\infty)}|V_i(\xi,t)|\le Ce^{-\mu_1 t}, \ t > 0,\ i = 1,2, \end{align*} |
for some
Proof. Denote
\begin{align*} z_{i}^{+}(t): = V_i(\infty,t),\ z_{i0}^{+}(s): = V_{i0}(\infty,s), \ s\in[-\tau,0], \ i = 1,2. \end{align*} |
Since
\lim\limits_{\xi\rightarrow+\infty}V_{i}(\xi,t) = z_{i}^{+}(t) |
exists uniformly for
\begin{align*} \begin{cases} \frac{d z_{1}^{+}}{dt}+\alpha z_{1}^{+}-h'(v_{2+})z_{2}^{+}(t-\tau) = P_1(z_{2}^{+}(t-\tau)),\\ \frac{d z_{2}^{+}}{dt}+\beta z_{2}^{+}-g'(v_{1+})z_{1}^{+}(t-\tau) = P_2(z_{1}^{+}(t-\tau)),\\ z_{i}^{+}(s) = z_{i0}^{+}(s), \ s\in[-\tau,0],\ i = 1,2, \end{cases} \end{align*} |
where
\begin{align*} \begin{cases} P_1(z_{2}^{+}) = h(v_{2+}+z_{2}^{+})-h(v_{2+})-h'(v_{2+})z_{2}^{+},\\ P_2(z_{1}^{+}) = g(v_{1+}+z_{1}^{+})-g(v_{1+})-g'(v_{1+})z_{1}^{+}. \end{cases} \end{align*} |
Then by [9,Lemma 3.8], there exist positive constants
\begin{align} |V_{i}(\infty,t)| = |z_{i}^{+}(t)|\leq Ce^{-\mu_{1}t},\ t > 0,\ i = 1,2, \end{align} | (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
\begin{align*} \widetilde{V}_i(\xi,t) = \sqrt{w(\xi)}V_i(\xi+x_0,t) = e^{-\lambda\xi}V_i(\xi+x_0,t),\; \; i = 1,2. \end{align*} |
Substituting
\begin{align} \begin{cases} \widetilde{V}_{1t}+c\widetilde{V}_{1\xi}+c_1\widetilde{V}_1(\xi,t) -d_1e^\lambda\widetilde{V}_1(\xi+1,t)-d_1e^{-\lambda}\widetilde{V}_1(\xi-1,t)\\ = \widetilde{Q}_1(\widetilde{V}_2(\xi-c\tau,t-\tau)),\\ \widetilde{V}_{2 t}+c\widetilde{V}_{2\xi}+c_2\widetilde{V}_2(\xi,t) -d_2e^\lambda\widetilde{V}_2(\xi+1,t)-d_2e^{-\lambda}\widetilde{V}_2(\xi-1,t)\\ = \widetilde{Q}_2(\widetilde{V}_1(\xi-c\tau,t-\tau)), \\ \widetilde{V}_i(\xi,s) = \sqrt{w(\xi)}V_{i0}(\xi+x_0,s) = :\widetilde{V}_{i0}(\xi,s), \ \xi\in\mathbb{R}, s\in[-\tau,0],\ i = 1,2, \end{cases} \end{align} | (18) |
where
\begin{align*} c_1 = c\lambda+2d_1+\alpha,\quad c_2 = c\lambda+2d_2+\beta \end{align*} |
and
\begin{align*} \widetilde{Q}_1(\widetilde{V}_2) = e^{-\lambda \xi}Q_1(V_2), \quad \widetilde{Q}_2(\widetilde{V}_1) = e^{-\lambda \xi}Q_2(V_1). \end{align*} |
By (11),
\begin{align} \widetilde{Q}_1(\widetilde{V}_2(\xi-c\tau,t-\tau)) = &e^{-\lambda \xi}Q_1(V_2(\xi-c\tau+x_0,t-\tau))\\ = &e^{-\lambda \xi}h'(\tilde\phi_2)V_2(\xi-c\tau+x_0,t-\tau)\\ = &e^{-\lambda c\tau}h'(\tilde\phi_2)\widetilde{V}_2(\xi-c\tau,t-\tau) \end{align} | (19) |
and
\begin{align} \widetilde{Q}_2(\widetilde{V}_1(\xi-c\tau,t-\tau)) = e^{-\lambda c\tau}g'(\tilde\phi_1)\widetilde{V}_1(\xi-c\tau,t-\tau). \end{align} | (20) |
By (H3), we further obtain
\begin{align*} |\widetilde{Q}_1(\widetilde{V}_2(\xi-c\tau,t-\tau))|\le h'(0) e^{-\lambda c\tau}|\widetilde{V}_2(\xi-c\tau,t-\tau)|,\\ |\widetilde{Q}_2(\widetilde{V}_1(\xi-c\tau,t-\tau))|\le g'(0) e^{-\lambda c\tau}|\widetilde{V}_1(\xi-c\tau,t-\tau)|. \end{align*} |
Taking (19) and (20) into (18), one can see that the coefficients
\begin{align} \begin{cases} V^+_{1 t}+c V^+_{1 \xi}+c_1V^+_1(\xi,t) -d_1e^\lambda V^+_1(\xi+1,t)-d_1e^{-\lambda}V^+_1(\xi-1,t)\\ \ = h'(0) e^{-\lambda c\tau}V^+_2(\xi-c\tau,t-\tau),\\ V^+_{2t}+cV^+_{2\xi}+c_2V^+_2(\xi,t) -d_2e^\lambda V^+_2(\xi+1,t)-d_2e^{-\lambda}V^+_2(\xi-1,t)\\ \ = g'(0) e^{-\lambda c\tau}V^+_1(\xi-c\tau,t-\tau), \end{cases} \end{align} | (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
\begin{align} \begin{cases} \frac{d}{dt}z(t) = Az(t)+Bz(t-\tau), \quad t\ge 0, \tau > 0, \\ z(s) = z_0(s),\quad s\in[-\tau,0]. \end{cases} \end{align} | (22) |
where
\begin{align*} z(t) = e^{A(t+\tau)}e_\tau^{B_1 t} z_0(-\tau)+\int_{-\tau}^0e^{A(t-s)} e_\tau^{B_1(t-\tau-s)} [z'_0(s)-Az_0(s)]ds, \end{align*} |
where
\begin{align*} e_\tau^{B_1 t} = \begin{cases} 0, &-\infty < t < -\tau,\\ I, &-\tau\le t < 0,\\ I+B_1 \frac{t}{1!}, &0\le t < \tau,\\ I+B_1 \frac{t}{1!}+ B_1^2\frac{ (t-\tau)^2}{2!}, &\tau\le t < 2\tau,\\ \vdots &\vdots\\ I+B_1\frac{t}{1!} + B_1^2\frac{ (t-\tau)^2}{2!}+\cdots+B_1^m\frac{ [t-(m-1)\tau]^m}{m!}, &(m-1)\tau\le t < m\tau,\\ \vdots &\vdots \end{cases} \end{align*} |
where
Lemma 3.4. ([11,Theorem 3.1]) Suppose
\begin{align*} \|z(t)\|\leq C_{0}e^{-\varepsilon_{\tau}\sigma t},\quad t > 0, \end{align*} |
where
\begin{align*} \|e^{At}e^{B_{1}t}_{\tau}\|\leq C_{0}e^{-\varepsilon_{\tau}\sigma t},\quad t > 0, \end{align*} |
where
From the proof of [11,Theome 3.1], one can see that
\begin{align*} \mu_1(A) = \lim\limits_{\theta\rightarrow 0^+}\frac{\|I+\theta A\|-1}{\theta} = \max\limits_{1\le j\le N}\left[Re(a_{jj})+\sum\limits_{j\not = i}^N|a_{ij}|\right] \end{align*} |
and
\begin{align*} \mu_\infty(A) = \lim\limits_{\theta\rightarrow 0^+}\frac{\|I+\theta A\|_\infty-1}{\theta} = \max\limits_{1\le i\le N}\left[Re(a_{ii})+\sum\limits_{i\not = j}^N|a_{ij}|\right]. \end{align*} |
Taking the Fourier transform to (21) and denoting the Fourier transform of
\begin{align} \begin{cases} \frac{\partial}{\partial t}\hat{V}^{+}_{1}(\eta,t) = \left(-c_1+d_1(e^{\lambda+\mathit{\boldsymbol{i}}\eta}+e^{-(\lambda+\mathit{\boldsymbol{i}}\eta)})-\mathit{\boldsymbol{i}}c\eta\right)\hat{V}^{+}_{1}(\eta,t) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +h'(0)e^{-c\tau(\lambda+\mathit{\boldsymbol{i}}\eta)}\hat{V}^{+}_{2}(\eta,t-\tau),\\ \frac{\partial}{\partial t}\hat{V}^{+}_{2}(\eta,t) = \left(-c_2+d_2(e^{\lambda+\mathit{\boldsymbol{i}}\eta}+e^{-(\lambda+\mathit{\boldsymbol{i}}\eta)})-\mathit{\boldsymbol{i}}c\eta\right)\hat{V}^{+}_{2}(\eta,t) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +g'(0)e^{-c\tau(\lambda+\mathit{\boldsymbol{i}}\eta)}\hat{V}^{+}_{1}(\eta,t-\tau),\\ \hat{V}^{+}_i(\eta,s) = \hat{V}^{+}_{i0}(\eta,s),\ \eta\in \mathbb{R},\ s\in[-\tau,0],\ i = 1,2. \end{cases} \end{align} | (23) |
Let
A(\eta) = \left( \begin{array}{cc} -c_1+d_1(e^{\lambda+\mathit{\boldsymbol{i}}\eta}+e^{-(\lambda+\mathit{\boldsymbol{i}}\eta)})-\mathit{\boldsymbol{i}}c\eta & 0 \\ 0 & -c_2+d_2(e^{\lambda+\mathit{\boldsymbol{i}}\eta}+e^{-(\lambda+\mathit{\boldsymbol{i}}\eta)})-\mathit{\boldsymbol{i}}c\eta \\ \end{array} \right) |
and
B(\eta) = \left( \begin{array}{cc} 0 & h'(0)e^{-c\tau(\lambda+\mathit{\boldsymbol{i}}\eta)} \\ g'(0)e^{-c\tau(\lambda+\mathit{\boldsymbol{i}}\eta)} & 0 \\ \end{array} \right). |
Then system (23) can be rewritten as
\begin{align} \hat{V}^{+}_{t}(\eta,t) = A(\eta)\hat{V}^{+}(\eta,t)+B(\eta)\hat{V}^{+}(\eta,t-\tau). \end{align} | (24) |
By Lemma 3.3, the linear delayed system (24) can be solved by
\begin{align} \hat{V}^{+}(\eta,t) = &e^{A(\eta)(t+\tau)}e^{B_1(\eta)t}_{\tau}\hat{V}_{0}^{+}(\eta,-\tau)\\ &+\int_{-\tau}^{0}e^{A(\eta)(t-s)}e^{B_1(\eta)(t-s-\tau)}_{\tau} \left[\partial_{s}\hat{V}_{0}^{+}(\eta,s)-A(\eta)\hat{V}_{0}^{+}(\eta,s)\right]ds\\ : = & I_{1}(\eta,t)+\int_{-\tau}^0I_{2}(\eta,t-s)ds, \end{align} | (25) |
where
\begin{align} &V^{+}(\xi,t) \end{align} | (26) |
\begin{align} & = \mathcal{F}^{-1}[I_1](\xi,t)+\int_{-\tau}^0\mathcal{F}^{-1}[I_{2}](\xi,t-s)ds\\ & = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{\mathit{\boldsymbol{i}}\xi\eta}e^{A(\eta)(t+\tau)} e^{B_1(\eta)t}_{\tau}\hat{V}_{0}^{+}(\eta,-\tau)d\eta\\ &\ \ \ +\frac{1}{2\pi}\int_{-\tau}^{0}\int_{-\infty}^{\infty}e^{\mathit{\boldsymbol{i}}\xi\eta}e^{A(\eta)(t-s)} e^{B_1(\eta)(t-s-\tau)}_{\tau}\left[\partial_{s}\hat{V}_{0}^{+}(\eta,s)-A(\eta)\hat{V}_{0}^{+}(\eta,s)\right]d\eta ds. \end{align} | (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
\begin{align*} \|V^{+}_{i}(t)\|_{L^{\infty}(\mathbb{R})}\leq Ce^{-\mu_{2}t}\ for \ c\ge \max\{c_*, c^*\},\ i = 1,2, \end{align*} |
where
Proof. According to (26), we shall estimate
\begin{align*} \mu(A(\eta)) = &\frac{\mu_{1}(A(\eta))+\mu_{\infty}(A(\eta))}{2}\\ = &\max\left\{-c_1+d_1(e^\lambda \cos\eta+e^{-\lambda}\cos\eta),-c_2+d_2(e^\lambda \cos\eta+e^{-\lambda}\cos\eta)\right\}\\ = &-c_2+d_2(e^\lambda \cos\eta+e^{-\lambda}\cos\eta)\\ = &-c_2+d_2(e^\lambda+e^{-\lambda})\cos\eta\\ = &-c\lambda+d_2(e^\lambda+e^{-\lambda}-2)-\beta-m(\eta), \end{align*} |
where
m(\eta) = d_2(1-\cos\eta)(e^\lambda+e^{-\lambda})\ge0, |
since
\begin{align*} \nu(B(\eta)) = \max\{h'(0), g'(0)\}e^{- \lambda c\tau}. \end{align*} |
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
\begin{align*} \left|\mu(A(\eta))\right|-\nu(B(\eta)) = &c\lambda-d_2(e^\lambda+e^{-\lambda}-2)+\beta+m(\eta) -\max\{h'(0), g'(0)\}e^{- \lambda c\tau} \notag\\ = &-\mathcal{P}_{2}(\lambda,c)+m(\eta), \end{align*} |
where
\begin{align} \|e^{A(\eta)(t+\tau)}e^{B_{1}(\eta)t}\|\leq C_{1}e^{-\varepsilon_{\tau}(|\mu(A(\eta))|-\nu(B(\eta)))t} \leq C_{1}e^{-\varepsilon_{\tau}\mu_{0}t}e^{-\varepsilon_{\tau}m(\eta)t}, \end{align} | (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
\begin{align} \sup\limits_{\xi\in\mathbb{R}}\|\mathcal{F}^{-1}[I_1](\xi,t)\| = &\sup\limits_{\xi\in\mathbb{R}}\left\|\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{\mathit{\boldsymbol{i}}\xi\eta}e^{A(\eta)(t+\tau)}e^{B_{1}(\eta)t}\hat{V}_{0}^{+}(\eta, -\tau)d\eta \right\|\\ \leq & C\int_{-\infty}^{\infty}e^{-\varepsilon_{\tau}m(\eta)t}e^{-\varepsilon_{\tau}\mu_{0}t} \|\hat{V}_{0}^{+}(\eta, -\tau)\|d\eta \\ \leq & Ce^{-\varepsilon_{\tau}\mu_{0}t}\sup\limits_{\eta\in\mathbb{R}}\|\hat{V}_{0}^{+}(\eta, -\tau)\| \int_{-\infty}^{\infty}e^{-\varepsilon_{\tau}m(\eta)t}d\eta \\ \leq & Ce^{-\mu_{2}t}\sum\limits_{i = 1}^2\|V_{i0}^{+}(\cdot, -\tau)\|_{L^{1}(\mathbb{R})}, \end{align} | (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
\begin{align*} &\sup\limits_{\xi\in\mathbb{R}}\|\mathcal{F}^{-1}[I_2](\xi,t-s)\|\notag \\& = \sup\limits_{\xi\in\mathbb{R}}\left\|\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{\mathit{\boldsymbol{i}}\xi\eta}e^{A(\eta)(t-s)}e^{B_{1}(\eta)(t-s-\tau)} \left[\partial_{s}\hat{V}_{0}^{+}(\eta, s)-A(\eta)\hat{V}_{0}^{+}(\eta, s)\right]d\eta\right\|\notag\\ &\leq C\int_{-\infty}^{\infty}e^{-\varepsilon_{\tau}m(\eta)(t-s)}e^{-\varepsilon_{\tau}\mu_{0}(t-s)} \left\|\partial_{s}\hat{V}_{0}^{+}(\eta, s)-A(\eta)\hat{V}_{0}^{+}(\eta, s)\right\|d\eta \notag\\ &\leq Ce^{-\varepsilon_{\tau}\mu_{0}t}e^{\varepsilon_{\tau}\mu_{0}s}\sup\limits_{\eta\in\mathbb{R}} \left\|\partial_{s}\hat{V}_{0}^{+}(\eta, s)-A(\eta)\hat{V}_{0}^{+}(\eta, s)\right\| \int_{-\infty}^{\infty}e^{-\varepsilon_{\tau}m(\eta)(t-s)}d\eta. \end{align*} |
It then follows that
\begin{align} &\int_{-\tau}^0\sup\limits_{\xi\in\mathbb{R}}\|\mathcal{F}^{-1}[I_2](\xi,t-s)\|ds\\ &\le Ce^{-\varepsilon_{\tau}\mu_{0}t} \int_{-\tau}^0e^{\varepsilon_{\tau}\mu_{0}s}\sup\limits_{\eta\in\mathbb{R}} \left\|\partial_{s}\hat{V}_{0}^{+}(\eta, s)-A(\eta)\hat{V}_{0}^{+}(\eta, s)\right\| \int_{-\infty}^{\infty}e^{-\varepsilon_{\tau}m(\eta)(t-s)}d\eta ds \\ &\le Ce^{-\varepsilon_{\tau}\mu_{0}t} \int_{-\tau}^0\|\partial_{s}V_{0}^{+}(\cdot,s)\|_{L^{1}(\mathbb{R})} +\|V_{0}^{+}(\cdot,s)\|_{W^{1,1}(\mathbb{R})}ds \\ &\le Ce^{-\varepsilon_{\tau}\mu_{0}t} \left(\|\partial_{s}V_{0}^{+}(s)\|_{L^1([-\tau, 0]; L^{1}(\mathbb{R}))} +\|V_{0}^{+}(s)\|_{L^1([-\tau, 0]; W^{1,1}(\mathbb{R}))}\right). \end{align} | (30) |
Substituting (29) and (30) to (26), we obtain the following the decay rate
\begin{align*} \sum\limits_{i = 1}^2\|V_{i}^{+}(t)\|_{L^{\infty}(\mathbb{R})}\leq Ce^{-\mu_{2}t}. \end{align*} |
This proof is complete.
The following maximum principle is needed to obtain the crucial boundedness estimate of
Lemma 3.6. Let
\begin{align} \begin{cases} \frac{\partial v}{\partial t}+a_1\frac{\partial v}{\partial \xi}+a_2 v -de^{\nu}v(t,\xi+1)-de^{-\nu}v(t,\xi-1)\ge 0,\ (t,\xi)\in (0,T]\times\mathbb{R},\\ v(0,\xi)\ge0, \quad \xi\in\mathbb{R}, \end{cases} \end{align} | (31) |
then
Lemma 3.7. When
Proof. When
\begin{align} h'(0)e^{-\lambda c\tau} V^+_2(\xi-c\tau,t-\tau) = h'(0)e^{-\lambda c\tau} V^+_{20}(\xi-c\tau,t-\tau)\ge 0. \end{align} | (32) |
Applying (32) to the first equation of (21), we get
\begin{align*} \begin{cases} V^+_{1 t}+c V^+_{1 \xi}+c_1V^+_1(\xi,t) -d_1e^\lambda V^+_1(\xi+1,t)-d_1e^{-\lambda}V^+_1(\xi-1,t)\\ \ge 0,\ (\xi,t)\in\mathbb{R}\times [0,\tau],\\ V^+_{10}(\xi,s)\ge 0, \ \xi\in\mathbb{R},\ s\in[-\tau,0]. \end{cases} \end{align*} |
By Lemma 3.6, we derive
\begin{align} V^+_1(\xi,t)\ge 0, \quad (\xi, t)\in \mathbb{R}\times[0,\tau]. \end{align} | (33) |
Similarly, we obtain
\begin{align*} \begin{cases} V^+_{2 t}+c V^+_{2\xi}+c_2V^+_2(\xi,t) -d_2e^\lambda V^+_2(\xi+1,t)-d_2e^{-\lambda}V^+_2(\xi-1,t)\\ \geq 0,\ (\xi,t)\in\mathbb{R}\times [0,\tau],\\ V^+_{20}(\xi,s)\ge 0, \ \xi\in\mathbb{R}\ s\in[-\tau,0]. \end{cases} \end{align*} |
Using Lemma 3.6 again, we obtain
\begin{align} V^+_2(\xi,t)\ge 0, \quad (\xi, t)\in \mathbb{R}\times[0,\tau]. \end{align} | (34) |
When
\begin{align} (V^+_1(\xi,t),V^+_2(\xi,t))\ge (0,0), \quad (\xi, t)\in \mathbb{R}\times [n\tau,(n +1)\tau]. \end{align} | (35) |
Combining (33), (34) and (31), we obtain
Now we establish the following crucial boundedness estimate for
Lemma 3.8. Let
\begin{align} |\widetilde{V}_{i0}(\xi,s)|\le V_{i0}^+(\xi,s) \quad \mathit{\mbox{for}}\quad(\xi,s)\in \mathbb{R}\times[-\tau, 0],\ i = 1,2, \end{align} | (36) |
then
\begin{align*} |\widetilde{V}_i(\xi,t)|\le V^+_i(\xi,t) \quad \mathit{\mbox{for}}\quad (\xi,t)\in \mathbb{R}\times[0, +\infty),\ i = 1,2. \end{align*} |
Proof. First of all, we prove
\begin{align} |\widetilde{V}_{i}(\xi-c\tau,t-\tau)|& = |\widetilde{V}_{i0}(\xi-c\tau,t-\tau)|\\ &\leq V_{i0}^{+}(\xi-c\tau,t-\tau)\\ & = V_{i}^{+}(\xi-c\tau,t-\tau) \quad \mbox{for}\ (\xi,t)\in\mathbb{R}\times [0,\tau]. \end{align} | (37) |
Then by
\begin{align} &h'(0)e^{-\lambda c\tau}V^{+}_2(\xi-c\tau,t-\tau)\pm h'(\tilde\phi_{2})e^{-\lambda c\tau}\widetilde{V}_2(\xi-c\tau,t-\tau)\\ &\geq h'(0)e^{-\lambda c\tau}V^{+}_2(\xi-c\tau,t-\tau) -|h'(\tilde\phi_{2})|e^{-\lambda c\tau}|\widetilde{V}_2(\xi-c\tau,t-\tau)|\\ &\geq 0 \quad \mbox{for}\ (\xi,t)\in\mathbb{R}\times [0,\tau] \end{align} | (38) |
and
\begin{align} &g'(0)e^{-\lambda c\tau}V^{+}_1(\xi-c\tau,t-\tau)\pm g'(\tilde\phi_{1})e^{-\lambda c\tau}\widetilde{V}_1(\xi-c\tau,t-\tau)\\ &\geq 0 \quad \mbox{for}\ (\xi,t)\in\mathbb{R}\times [0,\tau]. \end{align} | (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
\begin{align*} \begin{cases} U^-_{1t}+cU^-_{1\xi}+c_1U^-_1(\xi,t) -d_1e^\lambda U^-_1(\xi+1,t)-d_1e^{-\lambda}U^-_1(\xi-1,t)\\ \geq 0,\quad (\xi,t)\in\mathbb{R}\times [0,\tau],\\ U^{-}_{10}(\xi,s) = V^{+}_{10}(\xi,s)-\widetilde{V}_{10}(\xi,s)\geq 0, \quad \xi\in\mathbb{R},\ s\in[-\tau,0]. \end{cases} \end{align*} |
By Lemma 3.6, we obtain
U^{-}_{1}(\xi,t) \geq 0, \quad (\xi,t)\in \mathbb{R}\times[0,\tau], |
namely,
\begin{align} \widetilde{V}_{1}(\xi,t)\leq V_{1}^{+}(\xi,t),\quad (\xi,t)\in \mathbb{R}\times[0,\tau]. \end{align} | (40) |
Similarly, one has
\begin{align*} \begin{cases} U^-_{2t}+cU^-_{2\xi}+c_2U^-_2(\xi,t) -d_2e^\lambda U^-_2(\xi+1,t)-d_2e^{-\lambda}U^-_2(\xi-1,t)\\ \geq 0,\quad (\xi,t)\in\mathbb{R}\times [0,\tau],\\ U^{-}_{20}(\xi,s) = V^{+}_{20}(\xi,s)-\widetilde{V}_{20}(\xi,s)\geq 0, \quad \xi\in\mathbb{R},\ s\in[-\tau,0]. \end{cases} \end{align*} |
Applying Lemma 3.6 again, we have
U^{-}_{2}(\xi,t)\geq 0,\quad (\xi,t)\in\mathbb{R}\times[0,\tau], |
i.e.,
\begin{align} \widetilde{V}_{2}(\xi,t)\leq V_{2}^{+}(\xi,t),\quad (\xi,t)\in \mathbb{R}\times[0,\tau]{.} \end{align} | (41) |
On the other hand,
\begin{align*} \begin{cases} U^+_{1t}+cU^+_{1\xi}+c_1U^+_1(\xi,t) -d_1e^\lambda U^+_1(\xi+1,t)-d_1e^{-\lambda}U^+_1(\xi-1,t)\\ \geq 0,\quad (\xi,t)\in\mathbb{R}\times [0,\tau],\\ U^{-}_{10}(\xi,s) = V^{+}_{10}(\xi,s)-\widetilde{V}_{10}(\xi,s)\geq 0, \quad \xi\in\mathbb{R},\ s\in[-\tau,0]{.} \end{cases} \end{align*} |
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,
\begin{align} -V_{1}^{+}(\xi,t)\leq \widetilde{V}_{1}(\xi,t),\quad (\xi,t)\in \mathbb{R}\times[0,\tau]. \end{align} | (42) |
Similarly,
\begin{align*} \begin{cases} U^+_{2t}+c U^+_{2\xi}+c_2U^+_2(\xi,t) -d_2e^\lambda U^+_2(\xi+1,t)-d_2e^{-\lambda}U^+_2(\xi-1,t)\\ \geq 0,\quad (\xi,t)\in\mathbb{R}\times [0,\tau],\\ U^{-}_{20}(\xi,s) = V^{+}_{20}(\xi,s)-\widetilde{V}_{10}(\xi,s)\geq 0, \quad \xi\in\mathbb{R},\ s\in[-\tau,0]. \end{cases} \end{align*} |
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
\begin{align} -V_{2}^{+}(\xi,t)\leq \widetilde{V}_{2}(\xi,t),\quad (\xi,t)\in \mathbb{R}\times[0,\tau]. \end{align} | (43) |
Combining (40) and (42), we obtain
\begin{align} |\widetilde{V}_{1}(\xi,t)|\leq V^{+}_{1}(\xi,t)\quad \mbox{for}\quad(\xi,t)\in \mathbb{R}\times[0,\tau], \end{align} | (44) |
and combining (41) and (43), we prove
\begin{align} |\widetilde{V}_{2}(\xi,t)|\leq V^{+}_{2}(\xi,t)\quad \mbox{for}\quad(\xi,t)\in \mathbb{R}\times[0,\tau]. \end{align} | (45) |
Next, when
\begin{align*} |\widetilde{V}_{i}(\xi,t)| \leq V^{+}_{i}(\xi,t)\quad \mbox{for}\quad(\xi,t)\in \mathbb{R}\times[\tau,2\tau],\quad i = 1,2. \end{align*} |
Repeating this procedure, we then further prove
\begin{align*} |\widetilde{V}_{i}(\xi,t)| \leq V^{+}_{i}(\xi,t),\; \; (\xi,t)\in \mathbb{R}\times[n\tau,(n+1)\tau],\quad n = 1,2,\cdots, \end{align*} |
which implies
\begin{align*} |\widetilde{V}_{i}(\xi,t)| \leq V^{+}_{i}(\xi,t) \quad \mbox{for}\quad(\xi,t)\in \mathbb{R}\times[0,\infty), \quad i = 1,2. \end{align*} |
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
\begin{align*} \|\widetilde{V}_{i}(t)\|_{L^{\infty}(\mathbb{R})}\leq Ce^{-\mu_{2}t}, \end{align*} |
for some
Lemma 3.10. It holds that
\begin{eqnarray*} \sup\limits_{\xi\in(-\infty,x_{0}]}|V_{i}(\xi,t)|\leq Ce^{-\mu_{2}t},\ i = 1,2, \end{eqnarray*} |
for some
Proof. Since
\begin{align*} \sup\limits_{\xi\in(-\infty,0]}|V_{i}(\xi+x_{0},t)|\leq\|\widetilde{V}_{i}(t)\|_{L^{\infty}(\mathbb{R})}\leq Ce^{-\mu_{2}t}, \end{align*} |
which implies
\begin{align*} \sup\limits_{\xi\in(-\infty,x_0]}|V_{i}(\xi,t)|\leq Ce^{-\mu_{2}t}. \end{align*} |
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.
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