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

1.   Introduction

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

with the initial data

where $t>0$, $x\in\mathbb{R}$, $d_i\ge 0$ and

Here $v_1(x,t)$ and $v_2(x,t)$ biologically stand for the spatial density of the bacterial population and the infective human population at point $x\in\mathbb{R}$ and time $t\ge 0$, respectively. Both bacteria and humans are assumed to diffuse, $d_1$ and $d_2$ are diffusion coefficients; the term $-\alpha v_1$ is the natural death rate of the bacterial population and the nonlinearity $h(v_2)$ is the contribution of the infective humans to the growth rate of the bacterial; $-\beta v_2$ is the natural diminishing rate of the infective population due to the finite mean duration of the infectious population and the nonlinearity $g(v_1)$ 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 $\tau_1$, $\tau_2$ are time delays. The nonlinearities $g$ and $h$ satisfy the following hypothesis:

$\bf (H1)$ $g\in C^2([0,K_1], \mathbb{R})$, $g(0) = h(0) = 0$, $K_2 = g(K_1)/\beta>0$, $h\in C^2([0,K_2], \mathbb{R})$, $h(g(K_1)/\beta) = \alpha K_1$, $h(g(v)/\beta)>\alpha v$ for $v\in(0, K_1)$, where $K_1$ is a positive constant.

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

It is clear that (H1) is a basic assumption to ensure that system (1) is monostable on $[{\bf{0}}, {\bf{K}}]$. When $g'(u)\ge 0$ for $u\in[0,K_1]$ and $h'(v)\ge 0$ for $v\in [0, K_2]$, system (1) is a quasi-monotone system. Otherwise, if $g'(u)\ge 0$ for $u\in[0,K_1]$ or $h'(v)\ge 0$ for $v\in [0, K_2]$ 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 $(K_1, K_2)$. A traveling wave solution (in short, traveling wave) of (1) is a special translation invariant solution of the form $(v_1(x,t),v_2(x,t)) = (\phi_1(x+ct),\phi_2(x+ct))$, where $c>0$ is the wave speed. If $\phi_1$ and $\phi_2$ are monotone, then $(\phi_1, \phi_2)$ is called a traveling wavefront. Substituting $(\phi_1(x+ct),\phi_2(x+ct))$ into (1), we obtain the following wave profile system with the boundary conditions

where $\xi = x+ct$, $' = \frac{d}{d\xi}$, $\mathcal{D}[\phi_i](\xi) = \phi_{i}(\xi+1)-2\phi_{i}(\xi)+\phi_{i}(\xi-1)$, $i = 1,2$.

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

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

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.

2.   Preliminaries and main results

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 $\tau_1 = \tau_2 = \tau$.

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

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

can be expressed by

where $u_{0}(\cdot)\in L^{\infty}(\mathbb{R})$, $\mathbf{I}_{m}(\cdot)$, $m\geq0$ are defined as

and $\mathbf{I}_{m}(t) = \mathbf{I}_{-m}(t)$ for $m<0$. Moreover,

and $\mathbf{I}_{m}(0) = 0$ for $m\neq0$ while $\mathbf{I}_{0} = 1$, and $\mathbf{I}_{m}(t)\geq0$ for any $t\geq0$. In addition, one has

Thus, the solution $(v_1(x,t),v_2(x,t))$ of (6) can be expressed as

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

and

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

for $c\ge 0$ and $\lambda\in\mathbb{C}$, where

with

It is easy to see that $\Delta_1(c,\lambda) = 0$ admits two roots $\lambda_1^-<0<\lambda_1^+$, and $\Delta_2(c,\lambda) = 0$ has two roots $\lambda_2^-<0<\lambda_2^+$. We denote $\lambda_m^+ = \min\{\lambda_1^+,\lambda_2^+\}$.

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 $\mathcal{P}_1(c,\lambda) = 0$ has two distinct positive real roots $\lambda_1: = \lambda_1(c)$ and $\lambda_2: = \lambda_2(c)$ with $\lambda_1(c)< \lambda_2(c)<\lambda_m^+$, i.e. $\mathcal{P}_1(c,\lambda_1) = \mathcal{P}_1(c,\lambda_2) = 0$, and $\mathcal{P}(c,\lambda)>0$ for $\lambda\in(\lambda_1(c),\lambda_2(c))$. In addition, $\lim_{c\rightarrow c_*}\lambda_1(c) = \lim_{c\rightarrow c_*}\lambda_2(c) = \lambda_*>0$, i.e., $\mathcal{P}_1(c_*,\lambda_*) = 0$.

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:

$\bf (H2)$ There exist ${\bf{K}}^\pm = (K_1^\pm, K_2^\pm)\gg 0$ with ${\bf{K}}^-<{\bf{K}}<{\bf{K}}^+$ and four continuous and twice piecewise continuous differentiable functions $g^\pm:[0,K_1^+]\rightarrow \mathbb{R}$ and $h^\pm:[0,K_2^+]\rightarrow \mathbb{R}$ such that

(i) $K_2^\pm = g^\pm(K_1^\pm)/\beta$, $h^\pm(\frac{1}{\beta}g^\pm(K_1^\pm)) = \alpha K_1^\pm$, and $h^\pm(\frac{1}{\beta}g^\pm(v))>\alpha v$ for $v\in (0, K_1^\pm)$;

(ii) $g^\pm(u)$ and $h^\pm(v)$ are non-decreasing on $[0,K_1^+]$ and $[0,K_2^+]$, respectively;

(iii) $(g^\pm)'(0) = g'(0)$, $(h^\pm)'(0) = h'(0)$ and

Proposition 1. Assume that $\rm(H1)$ and $\rm(H2)$ hold, $\tau\ge 0$, and let $c_*$ be defined as in Lemma 2.1. Then for every $c>c_*$, system (1) has a traveling wave $(\phi_1(\xi),\phi_2(\xi))$ satisfying $(\phi_1(-\infty),\phi_2(-\infty)) = (0,0)$ and

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 $C_i (i = 1, 2,\dots)$ represents a specific constant. Let $\|\cdot\|$ and $\|\cdot\|_{\infty}$ denote $1$-norm and $\infty$-norm of the matrix (or vector), respectively. Let $I$ be an interval, typically $I = \mathbb{R}$. Denote by $L^1(I)$ the space of integrable functions defined on $I$, and $W^{k,1}(I)(k\ge0)$ the Sobolev space of the $L^1$-functions $f(x)$ defined on the interval $I$ whose derivatives $\frac{d^n }{dx^n}f(n = 1,\dots, k)$ also belong to $L^1(I)$. Let $L^1_ w(I)$ be the weighted $L^1$-space with a weight function $w(x) > 0$ and its norm is defined by

$W^{k,1}_w(I)$ be the weighted Sobolev space with the norm given by

Let $T> 0$ be a number and $\mathcal {B}$ be a Banach space. We denote by $C([0, T ]; \mathcal {B})$ the space of the $\mathcal {B}$-valued continuous functions on $[0, T ]$, and by $L^1([0, T ]; \mathcal {B})$ the space of the $\mathcal {B}$-valued $L^1$-functions on $[0, T ]$. The corresponding spaces of the $\mathcal {B}$-valued functions on $[0,\infty)$ are defined similarly. For any function $f(x)$, its Fourier transform is defined by

and the inverse Fourier transform is given by

where $\mathit{\boldsymbol{i}}$ is the imaginary unit, $\mathit{\boldsymbol{i}}^2 = -1$.

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

$\bf(H3)$ $|g'(u)|\le g'(0)$ and $|h'(v)|\le h'(0)$ for $u, v\in [0,+\infty)$.

$\bf(H4)$ $d_2> d_1$, $\alpha>\beta$, $d_2-d_1<\frac{\alpha-\beta}{2}$ and $\max\{h'(0), g'(0)\}>\beta$.

$\bf (H5)$ The initial data $(v_{10}(x,s), v_{20}(x,s))\ge (0,0)$ satisfies

Consider the following function

Since $\max\{h'(0), g'(0)\}>\beta$, it then follows from [20,Lemma 2.1] that there exists $\lambda^{*}>0$ and $c^{*}>0$, such that $\mathcal{P}_{2}(\lambda^{*},c^{*}) = 0$ and $\frac{\partial\mathcal{P}_{2}(\lambda,c)}{\partial\lambda}|_{(\lambda^{*},c^{*})} = 0$. When $c>c^*$, the equation $\mathcal{P}_{2}(\lambda,c) = 0$ has two positive real roots $\lambda^\natural_1(c)$ and $\lambda^\natural_2(c)$ with $0<\lambda^\natural_1(c)<\lambda^*< \lambda^\natural_2(c)$. When $\lambda\in (\lambda^\natural_1(c), \lambda^\natural_2(c))$, $\mathcal{P}_{2}(\lambda,c)<0$. Moreover, $(\lambda^\natural_1)'(c)<0$ and $(\lambda^\natural_2)'(c)>0$.

We select the weight function $w(\xi) > 0$ as the form

where $\lambda>0$ satisfies $\lambda^\natural_1(c)<\lambda<\lambda^\natural_2(c)$. Now we are ready to present the main result of this paper.

Theorem 2.2. (Global stability of traveling waves). Assume that $\rm(H1)$, $\rm(H3)$-$\rm(H5)$ hold. For any given traveling wave $(\phi_1(x + ct),\phi_2(x+ct))$ of (1) with speed $c> \max\{c_*, c^*\}$ connecting $(0,0)$ and $(K_1,K_2)$, whether it is monotone or non-monotone, if the initial data satisfy

then there exists $\tau_0>0$ such that for any $\tau\le \tau_0$, the solution $(v_1(x,t), v_2(x,t))$ of (1)-(2) converges to the traveling wave $(\phi_1(x+ct),\phi_2(x+ct))$ as follows:

where $C$ and $\mu$ are two positive constants, and $C_{unif}[r,T]$ is the uniformly continuous space, for $0<T\le\infty$, defined by

3.   Global stability of traveling waves

This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let $(\phi_1(x+ct),\phi_2(x+ct)) = (\phi_1(\xi), \phi_2(\xi))$ be a given traveling wave solution with speed $c\ge c_*$ and define

Then it follows from (1) and (3) that $V_i(\xi,t)$ satisfies

The nonlinear terms $Q_1$ and $Q_2$ are given by

for some $\tilde \phi_i$ between $\phi_i$ and $\phi_i+V_i$, with $\phi_i = \phi_i(\xi-c\tau_i)$ and $V_i = V_i(\xi-c\tau_i, t-\tau_i)$.

We first prove the existence and uniqueness of solution $(V_1(\xi,t), V_2(\xi,t))$ to the initial value problem (10) in the uniformly continuous space $C_{unif}[-\tau, +\infty)\times C_{unif}[-\tau, +\infty)$.

Lemma 3.1. Assume that $\rm(H1) and (H3)$ hold. If the initial perturbation $(V_{10}, V_{20}) \in C_{unif}[-\tau, 0]\times C_{unif}[-\tau, 0]$ for $c\ge c_*$, then the solution $(V_1,V_2)$ of the perturbed equation (10) is unique and time-globally exists in $C_{unif}[-\tau, +\infty)\times C_{unif}[-\tau, +\infty)$.

Proof. Let $U_{i}(x,t) = v_{i}(x,t)-\phi_{i}(x+ct)$, $i = 1,2$. It is clear that $U_{i}(x,t) = V_{i}(\xi,t)$, $i = 1,2$, and satisfies

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

When $t\in [0,\tau]$, we have $t-\tau\in [-\tau,0]$ and $U_{i}(x,t-\tau) = U_{i0}(x,t-\tau)$, $i = 1,2$, which imply that (12) is linear. Thus, the solution of (12) can be explicitly and uniquely solved by

for $t\in [0,\tau]$.

Since $V_{i0}(\xi,t)\in C_{unif}[-\tau, 0]$, $i = 1,2$, namely, $\lim\limits_{\xi\rightarrow+\infty}V_{i0}(\xi,t)$ exist uniformly in $t\in [-\tau,0]$, which implies $\lim\limits_{x\rightarrow+\infty}U_{i0}(x,t)$ exist uniformly in $t\in [-\tau,0]$. Denote $U_{i0}(\infty,t) = \lim\limits_{x\rightarrow+\infty}U_{i0}(x,t)$, $i = 1,2$. Taking the limit $x\rightarrow+\infty$ to (13) yields

and

where we have used (8). Thus, we obtain that $(U_{1},U_{2})\in C_{unif}[-\tau, \tau)\times C_{unif}[-\tau, \tau)$.

When $t\in[\tau, 2\tau]$, system (12) with the initial data $U_{i}(x,s)$ for $s\in[0, \tau]$ is still linear, because the source term $Q_1(U_{2}(x,t-\tau))$ and $Q_2(U_{1}(x, t-\tau))$ is known due to $t-\tau\in[0,\tau]$ and $U_{i}(s,t-\tau)$ is solved in (13). Hence, the solution $U_i(x,t)$ for $t\in[\tau,2\tau]$ is uniquely and explicitly given by

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

and

By repeating this procedure for $t\in[n\tau,(n+1)\tau]$ with $n\in \mathbb{Z}_{+}$, we prove that there exists a unique solution $(V_{1},V_{2})\in C_{unif}[-\tau,(n+1)\tau]\times C_{unif}[-\tau,(n+1)\tau]$ for (10), and step by step, we finally prove the uniqueness and time-global existence of the solution $(V_{1},V_{2})\in C_{unif}[-\tau,\infty)\times C_{unif}[-\tau,\infty)$ 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 $\rm(H1)$, $\rm(H3)$-$\rm(H5)$ hold. If

and

then there exists $\tau_0>0$ such that for any $\tau\le\tau_0$, when $c>\max\{c_*, c^*\}$, it holds

for some $\mu>0$ and $C>0$.

In order to prove Proposition 2, we first investigate the decay estimate of $V_i(\xi,t)$ at $\xi = +\infty$, $i = 1, 2$.

Lemma 3.2. Assume that $V_{i0}\in C_{unif}[-\tau,0]$, $i = 1,2$. Then, there exist $\tau_0>0$ and a large number $x_0\gg 1$ such that when $\tau\le \tau_0$, the solution $V_i(\xi, t)$ of (10) satisfies

for some $\mu_1>0$ and $C>0$.

Proof. Denote

Since $V_{i0}\in C_{unif}[-\tau,0]$, $i = 1,2$, by Lemma 3.1, we have $V_{i}\in C_{unif}[-\tau,+\infty)$, which implies

exists uniformly for $t\in[-\tau,+\infty)$. Taking the limit $\xi\rightarrow+\infty$ to (10), we obtain

where

Then by [9,Lemma 3.8], there exist positive constants $\tau_0$, $\mu_1$ and $C$ such that when $\tau\le \tau_0$,

provided that $|z_{i0}^{+}|\ll 1$, $i = 1,2.$

By the continuity and the uniform convergence of $V_{i}(\xi,t)$ as $\xi\rightarrow+\infty$, there exists a large $x_{0}\gg 1$ such that (17) implies

provided that $\sup\limits_{\xi\in[x_{0},+\infty)}|V_{i0}(\xi,s)|\ll 1$ for $s\in[-\tau,0]$. Such a smallness for the initial perturbation $(V_{10},V_{20})$ near $\xi\rightarrow+\infty$ can be easily verified, since

which implies

uniformly for $s\in[-\tau,0]$, $i = 1,2$. The proof is complete.

Next we are going to establish the a priori decay estimate of $\sup_{\xi\in(-\infty, x_0]}|V_i(\xi,t)|$ by using the anti-weighted technique [3] together with the Fourier transform. First of all, we shift $V_i(\xi,t)$ to $V_i(\xi+x_0,t)$ by the constant $x_0$ given in Lemma 3.2, and then introduce the following transformation

Substituting $V_i = w^{-1/2}\widetilde{V}_i$ to (10) yields

where

and

By (11), $\tilde Q_1(\tilde{V}_2)$ satisfies

and $\widetilde{Q}_2(\widetilde{V}_1)$ satisfies

By (H3), we further obtain

Taking (19) and (20) into (18), one can see that the coefficients $h'(\tilde\phi_2)$ and $g'(\tilde\phi_1)$ 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 $(\widetilde{V}_1, \widetilde{V}_2)$. Motivated by [15,28,17,23], we introduce a new method which can be described as follows.

$\circ$ By replacing $h'(\tilde \phi_2)$ in the first equation of (18) with a constant $h'(0)$, and $g'(\tilde \phi_1)$ in the second equation of (18) with a constant $g'(0)$, we can obtain a linear delayed reaction-diffusion system

with

where $\xi\in\mathbb{R}$, $t\in(0,+\infty]$ and $s\in[-\tau,0]$. Then we investigate the decay estimate of $(V_1^+,V_2^+)$ by applying the Fourier transform to (21);

$\circ$ We prove that the solution $(\widetilde{V}_1,\widetilde{V}_2)$ of (18) can be bounded by the solution $(V_1^+,V_2^+)$ of (21).

Now we are in a position to derive the decay estimate of $(V_1^+,V_2^+)$ 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

where $A, B\in \mathbb{C}^{N\times N}$, $N\ge 2$, and $z_0(s)\in C^1([-\tau, 0], \mathbb{C}^N)$. Then

where $B_1 = Be^{-A\tau}$ and $e_\tau^{B_1 t}$ is the so-called delayed exponential function in the form

where $0, I\in \mathbb{C}^{N\times N}$, and $0$ is zero matrix and $I$ is unit matrix.

Lemma 3.4. ([11,Theorem 3.1]) Suppose $\mu(A): = \frac{\mu_{1}(A)+\mu_{\infty}(A)}{2}<0$, where $\mu_{1}(A)$ and $\mu_{\infty}(A)$ denote the matrix measure of $A$ induced by the matrix $1$-norm $\|\cdot\|_1$ and $\infty$-norm $\|\cdot\|_{\infty}$, respectively. If $\nu(B): = \frac{\|B\|+\|B\|_{\infty}}{2}\leq -\mu(A)$, then there exists a decreasing function $\varepsilon_{\tau} = \varepsilon(\tau)\in(0,1)$ for $\tau > 0$ such that any solution of system (22) satisfies

where $C_{0}$ is a positive constant depending on initial data $z_{0}(s),s\in [-\tau,0]$ and $\sigma = |\mu(A)|-\nu(B)$. In particular,

where $e_\tau^{B_1 t}$ is defined in Lemma 3.3.

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

and

Taking the Fourier transform to (21) and denoting the Fourier transform of $V^{+}(\xi,t): = (V^{+}_{1}(\xi,t),V^{+}_{2}(\xi,t))^{T}$ by $\hat{V}^{+}(\eta,t): = (\hat{V}^{+}_{1}(\eta,t),\hat{V}^{+}_{2}(\eta,t))^{T}$, we obtain

Let

and

Then system (23) can be rewritten as

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

where $B_1(\eta) = B(\eta)e^{A(\eta)\tau}$. Then by taking the inverse Fourier transform to (25), one has

Lemma 3.5. Let the initial data $V_{i0}^{+}(\xi,s)$, $i = 1,2$, be such that

Then

where $\mu_{2}>0$ and $C>0$.

Proof. According to (26), we shall estimate $\mathcal{F}^{-1}[I_1](\xi,t)$ and $\int_{-\tau}^0\mathcal{F}^{-1}[I_{2}](\xi,t-s)ds$, respectively. By the definition of $\mu(\cdot)$ and $\nu(\cdot)$, we have

where $c_2 = c\lambda+2d_2+\beta$ and

since $d_2> d_1$, $\alpha>\beta$ and $d_2-d_1<\frac{\alpha-\beta}{2}$, and

By considering $\lambda\in(\lambda_1^\natural(c), \lambda_2^\natural(c))$, we get $\mu(A(\eta))<0$ and

Furthermore, we obtain

where $\mathcal{P}_{2}(\lambda,c) = d_2(e^\lambda+e^{-\lambda}-2)-c\lambda-\beta+\max\{h'(0), g'(0)\}e^{- \lambda c\tau}<0$ for $c>\max\{c_*, c^*\}$. It then follows from Lemma 3.4 that there exists a decreasing function $\varepsilon_{\tau} = \varepsilon(\tau)\in(0,1)$ such that

where $C_{1}$ is a positive constant and $\mu_{0}: = -\mathcal{P}_{2}(\lambda,c)>0$ with $c>c^*$. By the definition of Fourier's transform, we have

Applying (28), we derive

with $\mu_{2}: = \varepsilon_{\tau}\mu_{0}$.

Note that

Similarly, we can obtain

It then follows that

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

This proof is complete.

The following maximum principle is needed to obtain the crucial boundedness estimate of $(\widetilde{V}_1,\widetilde{V}_2)$, which has been proved in [17,Lemma 3.4].

Lemma 3.6. Let $T>0$. For any $a_1, a_2 \in\mathbb{R}$ and $\nu>0$, if the bounded function $v$ satisfies

then $v(t,\xi)\ge 0$ for all $(t,\xi)\in (0,T]\times\mathbb{R}$.

Lemma 3.7. When $(V^+_{10}(\xi,s),V^+_{20}(\xi,s))\ge (0,0)$ for $(\xi,s)\in \mathbb{R}\times [-\tau,0]$, then $(V_1^+(\xi,t),V_2^+(\xi,t))\ge(0,0)$ for $(\xi,t)\in \mathbb{R}\times[0, +\infty)$.

Proof. When $t\in [0,\tau]$, we have $t-\tau\in[-\tau,0]$ and

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

By Lemma 3.6, we derive

Similarly, we obtain

Using Lemma 3.6 again, we obtain

When $t\in [n\tau,(n +1)\tau]$, $n = 1 ,2, \cdots$, repeating the above procedure step by step, we can similarly prove

Combining (33), (34) and (31), we obtain $(V^+_1(\xi,t),V^+_2(\xi,t))\ge (0,0)$ for $(\xi,t)\in \mathbb{R}\times [0,+\infty)$. The proof is complete.

Now we establish the following crucial boundedness estimate for $(\widetilde{V}_1, \widetilde{V}_2)$.

Lemma 3.8. Let $(\widetilde{V}_1(\xi,t), \widetilde{V}_2(\xi,t))$ and $(V^+_1(\xi,t),V^+_2(\xi,t))$ be the solutions of (18) and (21), respectively. When

then

Proof. First of all, we prove $|\widetilde{V}_i(\xi,t)|\leq V^+_i(\xi,t)$ for $t\in[0,\tau],i = 1,2.$ In fact, when $t\in[0,\tau]$, namely, $t-\tau\in[-\tau,0]$, it follows from (36) that

Then by $|h'(\tilde\phi_{2})|<h'(0)$ and $|g'(\tilde\phi_{1})|< g'(0)$ and (37), we get

and

Let

We are going to estimate $U_{i}^{\pm}(\xi,t)$ respectively.

From (18), (19), (21) and (38), we see that $U_{1}^{-}(\xi,t)$ satisfies

By Lemma 3.6, we obtain

namely,

Similarly, one has

Applying Lemma 3.6 again, we have

i.e.,

On the other hand, $U^{+}_{1}(\xi,t)$ satisfies

Then Lemma 3.6 implies that

that is,

Similarly, $U^{+}_{2}(\xi,t)$ satisfies

Therefore, we can prove that

namely

Combining (40) and (42), we obtain

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

Next, when $t\in[\tau,2\tau]$, namely, $t-\tau\in[0,\tau]$, based on (44) and (45), we can similarly prove

Repeating this procedure, we then further prove

which implies

The proof is complete.

Let us choose $V_{i0}^{+}(\xi,s)$ such that

and

Combining Lemmas 3.5 and 3.8, we can get the convergence rates for $\widetilde{V}(\xi,t).$

Lemma 3.9. When $\widetilde{V}_{i0}\in C([-\tau,0];W^{1,1}(\mathbb{R}))$ and $\partial_{s}\widetilde{V}_{i0}\in L^1([-\tau,0]; L^{1}(\mathbb{R}))$, then

for some $\mu_{2}>0$, $i = 1,2$.

Lemma 3.10. It holds that

for some $\mu_{2}>0$.

Proof. Since $\widetilde{V}_{i}(\xi,t) = \sqrt{w(\xi)}V_{i}(\xi+x_{0},t) = e^{-\lambda\xi}V_{i}(\xi+x_{0},t)$ and $\sqrt{w(\xi)} = e^{-\lambda\xi}\geq 1$ for $\xi\in (-\infty,0]$, then we obtain

which implies

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

Proof of Proposition 3.2. By Lemmas 3.2 and 3.10, we immediately obtain (16) for $0<\mu<\min\{\mu_{1},\mu_{2}\}.$

Acknowledgments

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