This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in with the exponential convergence rate for some constant .
Citation: Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay[J]. Electronic Research Archive, 2021, 29(4): 2599-2618. doi: 10.3934/era.2021003
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This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in with the exponential convergence rate for some constant .
In this article, we consider the following spatially discrete diffusion system with time delay
(1) |
with the initial data
(2) |
where
Here
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
(3) |
where
System (1) is a discrete version of classical epidemic model
(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
(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.,
(6) |
According to [8], with aid of modified Bessel functions, the solution to the initial value problem
can be expressed by
where
and
(7) |
and
(8) |
Thus, the solution
(9) |
In fact, by [8,Lemma 2.1], we can differentiate the series on
and
Next we investigate the characteristic roots of the linearized system for the wave profile system (3) at the trivial equilibrium
for
with
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)
Proposition 1. Assume that
Finally, we shall state the stability result of traveling waves derived in Proposition 1. Before that, let us introduce the following notations.
Notations.
Let
and the inverse Fourier transform is given by
where
To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.
Consider the following function
Since
We select the weight function
where
Theorem 2.2. (Global stability of traveling waves). Assume that
then there exists
where
This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let
Then it follows from (1) and (3) that
(10) |
The nonlinear terms
(11) |
for some
We first prove the existence and uniqueness of solution
Lemma 3.1. Assume that
Proof. Let
(12) |
Thus, the global existence and uniqueness of solution of (10) are transformed into that of (12).
When
(13) |
for
Since
(14) |
and
(15) |
where we have used (8). Thus, we obtain that
When
Similarly, by (14) and (15), we have
and
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
and
then there exists
(16) |
for some
In order to prove Proposition 2, we first investigate the decay estimate of
Lemma 3.2. Assume that
for some
Proof. Denote
Since
exists uniformly for
where
Then by [9,Lemma 3.8], there exist positive constants
(17) |
provided that
By the continuity and the uniform convergence of
provided that
which implies
uniformly for
Next we are going to establish the a priori decay estimate of
Substituting
(18) |
where
and
By (11),
(19) |
and
(20) |
By (H3), we further obtain
Taking (19) and (20) into (18), one can see that the coefficients
(21) |
with
where
Now we are in a position to derive the decay estimate of
Lemma 3.3. ([11,Lemma 3.1]) Let
(22) |
where
where
where
Lemma 3.4. ([11,Theorem 3.1]) Suppose
where
where
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
(23) |
Let
and
Then system (23) can be rewritten as
(24) |
By Lemma 3.3, the linear delayed system (24) can be solved by
(25) |
where
(26) |
(27) |
Lemma 3.5. Let the initial data
Then
where
Proof. According to (26), we shall estimate
where
since
By considering
Furthermore, we obtain
where
(28) |
where
Applying (28), we derive
(29) |
with
Note that
Similarly, we can obtain
It then follows that
(30) |
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
Lemma 3.6. Let
(31) |
then
Lemma 3.7. When
Proof. When
(32) |
Applying (32) to the first equation of (21), we get
By Lemma 3.6, we derive
(33) |
Similarly, we obtain
Using Lemma 3.6 again, we obtain
(34) |
When
(35) |
Combining (33), (34) and (31), we obtain
Now we establish the following crucial boundedness estimate for
Lemma 3.8. Let
(36) |
then
Proof. First of all, we prove
(37) |
Then by
(38) |
and
(39) |
Let
We are going to estimate
From (18), (19), (21) and (38), we see that
By Lemma 3.6, we obtain
namely,
(40) |
Similarly, one has
Applying Lemma 3.6 again, we have
i.e.,
(41) |
On the other hand,
Then Lemma 3.6 implies that
that is,
(42) |
Similarly,
Therefore, we can prove that
namely
(43) |
Combining (40) and (42), we obtain
(44) |
and combining (41) and (43), we prove
(45) |
Next, when
Repeating this procedure, we then further prove
which implies
The proof is complete.
Let us choose
and
Combining Lemmas 3.5 and 3.8, we can get the convergence rates for
Lemma 3.9. When
for some
Lemma 3.10. It holds that
for some
Proof. Since
which implies
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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