Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 6910491, 13 pages
https://doi.org/10.1155/2018/6910491

Research Article

Global Stability of Traveling Waves for a More General Nonlocal Reaction-Diffusion Equation

Rui Yan¹ and Guirong Liu²

¹School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030006, China
²School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Guirong Liu; lgr5791@sxu.edu.cn

Received 11 February 2018; Accepted 20 May 2018; Published 27 June 2018

Academic Editor: Pilar R. Gordoa

Copyright © 2018 Rui Yan and Guirong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to investigate the global stability of traveling front solutions with noncritical and critical speeds for a more general nonlocal reaction-diffusion equation with or without delay. Our analysis relies on the technical weighted energy method and Fourier transform. Moreover, we can get the rates of convergence and the effect of time-delay on the decay rates of the solutions. Furthermore, according to the stability results, the uniqueness of the traveling front solutions can be proved. Our results generalize and improve the existing results.

1. Introduction

In the study of biology and other subject fields, the reaction-diffusion equations with delays are usually utilized to depict the population distribution and physical evolution process and so forth, for instance, [1–6]. In this paper, we will study a more general reaction-diffusion equation

with the initial data

where is the Laplacian operator on , . Some mathematical models in the literature can be depicted by (1) with proper selections of , and . For instance, setting

(1) turns to the following famous nonlocal Nicholson’s blowflies population model

where is often formed as

in (4). Particularly, if , (4) is Nicholson’s birth rate function. Furthermore, setting

(1) turns to the classical Fisher-KPP equation as follows:

Now we impose some assumptions on (1) as follows:

() is a continuous nonnegative function with and

() Let , , , and for any . In addition, for any , where and .() , , and , , for any .() .

From , it is easy to see that , are two constant equilibria . Throughout this paper, a traveling front solution of (1) connecting is a nondecreasing solution with the form ; that is, it satisfies the following ordinary differential equation:

where is the traveling wave speed.

In the past few years, the study on traveling waves of the reaction-diffusion equations has drawn wide attention. One of the important and difficult problems is the stability of traveling waves. For example, the authors in [7, 8] and the references therein proved the stability of traveling waves of reaction-diffusion equations without time-delay. In fact, for the time-delayed reaction-diffusion equations, Schaaf [1] firstly studied the stability of the traveling waves by applying a spectral analysis. Later, there are many great contributions on this issue on both time-delayed and nonlocal reaction-diffusion equations. For instance, in [9] the authors investigated that the traveling front solutions with noncritical speeds were globally asymptotically stable by using the super- and subsolution method. Though the study of the stability of traveling waves in monostable condition is difficult, Mei in [10] firstly showed nonlinear stability of the traveling front solutions of a time-delayed diffusive Nicholson blowflies equation by employing a technical weighted energy method. Then Mei and coauthors in [11–15] further obtained global stability using both the weighted method and the comparison principle. Among them, the authors in [11] developed and improved the wave stability results showed in [10]. By using the above methods, Wu et al. in [16] showed the exponential stability of traveling wavefronts in monostable reaction-advection-diffusion equations with nonlocal delay, which improved some previous works. In a word, there are three commonly used methods for proving the stability of traveling waves, which we mentioned above.

The most challenging problem, however, is the stability of the critical traveling wave solutions to local or nonlocal time-delayed equations. It is also very important because the critical wave speed is the spreading speed. The methods mentioned above can not be used to solve this problem. As a matter of fact, as early as in 1978, by using the maximum principle method, Uchiyama [17] gave the local stability of the traveling waves including the critical waves (no convergence rate). Immediately, Moet [18] proved that the critical waves of the KPP equation were algebraically stable by using the Green function method. Later, Kirchgässner [19] and Gallay [20] showed the stability of the critical waves by using the spectral method and the renormalization group method for parabolic equations, respectively. Recently, for some nonlocal time-delayed reaction-diffusion equations, Mei, Ou and Zhao [21] and Wang [22] proved the globally exponential stability of traveling front solutions with noncritical speeds and globally algebraical stability of traveling front solutions with critical speed by using the weighted energy method and Green’s function method. Particularly, Mei and Wang [23] considered a class of nonlocal time-delayed Fisher-KPP type reaction-diffusion equations in -dimensional space. They obtained the exponential stability of all noncritical planar wavefronts and the algebraic stability of the critical planar wavefronts by using the weighted energy method coupled with Fourier transform. Furthermore, the convergence rates were obtained in the sense with -initial perturbation. Very recently, Chern et al. [24] studied the stability of critical traveling waves for a kind of nonmonotone time-delayed reaction-diffusion equations by using the technical weighted energy method with some new developments.

The main purpose of this paper is to investigate the stability of the traveling front solutions of (1) including the traveling waves with critical speed. First, let us review the works of the existence of the traveling front solutions of (1). In [25], Wang showed the existence of traveling front solutions with speed for (1) with nonmonotone nonlinearity by constructing a closed and convex subset in a suitable Banach space and using the fixed point theorem, where is the minimal wave speed. Recently, for the reaction-diffusion equations with nonlocal delays, Tian [26] proved the existence of the traveling waves with by using the finite time-delay approximation method coupled with the monotone semiflows theorem. Then in this paper, by using the technical weighted energy method and Fourier transform, we obtained the exponential stability of traveling front solutions with noncritical speeds and the algebraic stability of the traveling front solutions with critical speed of (1). Furthermore, the convergence rates and the effect of time-delay on the decay rates of the solutions were showed. At last, motivated by Lin, Lin, and Mei [27], we show the uniqueness of traveling front solutions for (1).

The rest of the paper is organized as follows. In Section 2, we introduce some preliminaries used later. In Section 3, we will prove the stability of the traveling front solutions for (1) by using the weighted energy method and Fourier transform. According to the stability results, in Section 4 the uniqueness of the traveling front solutions can be proved. In the last section, we apply our results to some models.

2. Preliminaries

In this section, we introduce some notations as follows. We assume that represents a general constant and denotes a concrete constant. Set to be an interval, ordinarily . Take

with the norm defined by

Moreover, is the Sobolev space with the norm given by

In addition, we assume that denotes a positive number and represents a Banach space. Also we let be the space of the -valued continuous functions on. Similarly, we can define the corresponding spaces of -valued functions on .

Next we present some previous results which will be needed in the proofs of our results later.

Lemma 1 (see [28]). Set to be the solution to the following linear time-delayed ODE with time-delay

Thus

where

and is the delayed exponential function defined by

and is the fundamental solution as

Lemma 2 (see [23]). Set and . Thus the solution of (13) satisfies

where

and the fundamental solution with of (17) satisfies

for arbitrary constant .
Moreover, if , there is a number such that

and the solution of (13) satisfies

where is uniquely determined by

Moreover, clearly, conditions guarantees the existence of the traveling front solutions of (1) which was showed in [25, 26, 29]. So we have the following result.

Theorem 3 (existence of traveling waves). Suppose that hold, there is a pair determined by

where

For any , (1) has a nondecreasing traveling front solution such that and , while for any , (1) has no traveling front solution connecting and . Moreover, when , the traveling front solutions with noncritical speeds satisfy with some positive constant .
Furthermore, when , has two distinct positive real roots and with such that

When , it holds that

Now we define a weight function with a number ,

with a sufficiently large number , where when , but when . Obviously for all and as .

3. The Stability of Traveling Front Solutions

In this section, we will prove the stability of all traveling front solutions with time-delay or not. Firstly, we show the following boundedness and establish the comparison principle for (1). Here we omit the proofs of these results since it is essentially the same as that of [14].

Lemma 4 (boundedness). Assume hold and the initial data satisfy

then the solution of the Cauchy problem (1) and (2) exists uniquely and satisfies

Lemma 5 (comparison principle). Set and to be the solution of (1) and (2) with the initial data and , respectively. If

thus

For the given , if

set

thus

Let and be the corresponding solutions of (1) and (2) with initial data and defined in (34) respectively; that is,

By Lemma 5, we can get

Now we need the following three steps.

Firstly, we will prove the convergence of to

For any , set and

From (36) and (39) we get

Then it can be verified that defined in (40) satisfies

with the initial data

where

and

By using the mollification, Zorns lemma, and energy method, we obtain the global existence for the solution of (42)-(43).

For the nonlinearity , applying Taylor’s formula to (45) and noting , we have

where is some function between and ; is some function between and . Then (42) becomes

Set to be the solution of the following equation with the original data :

By the assumptions and the comparison principle, we see that

Take

then satisfies the following equation:

where

When , by taking Fourier transform

to (51), we have

where

According to Lemma 1, we get the solution of (54) as

where

Next by taking the inverse Fourier transform to (57), we get

where the inverse Fourier transform is given by

Now we will prove the asymptotic behavior of .

Lemma 6 (decay rates for ). Suppose and . Thus, when , there is a number determined in Lemma 2 such that the solution of (51) satisfies

Proof. Set

and

It follows from (55) and (56) that

and

Because of , we can get

By (62) and combining with (65)-(68), we have

Let . Noticing for and , from Lemma 2, it holds that

where . Then,

On the other hand, by applying the characters of Fourier transform, there holds

and

Then, by a direct calculation, we obtain

and

From (71) and (75), we can get the result.

Then, for traveling front solutions with noncritical speeds, from Lemma 6 we get the following exponential decay:

But if , from (25) and (27), we see that

Then, for traveling front solutions with critical speeds, from Lemma 6 we get the following algebraic decay:

Since for , we can directly get the following result for .

Lemma 7. There holds that

and

where .

Moreover, we need to prove the decay rate of for .

Lemma 8. There holds that

with some constant which satisfies

and

where , , and is the root of the equation .

Proof. Now we consider the following equations, for :

and for ,

with two constants and .
When , take

where is a large number. Now we can choose and such that for , ,

Thus, we can verify that

On the other hand, for the interval , by choosing large enough, we can obtain ; that is, is an upper solution of (84). Therefore, for , we obtain

When , set

where is a large number. Similarly, we can obtain

Therefore, for , we obtain

This completes the proof.

From Lemmas 7 and 8, we prove the convergence directly as follows.

Lemma 9. There holds that
(1) if , one gets

with some sufficiently small constant ;
(2) if , one gets

Secondly, we will prove the convergence of to

As shown in the above process, we can similarly prove the following convergence of to .

Lemma 10. There holds that
(1) if , one can get

with some sufficiently small constant ;
(2) if , one can get

Lastly, we will prove the convergence of to

Lemma 11. There holds that:
(1) if , one can get

with some sufficiently small constant ;
(2) if , one can get

Combining with the above lemmas, we can get the following stability results of the traveling front solutions for (1).

Theorem 12 (stability of traveling waves with time-delay). Suppose hold. When , for the given traveling front solution of (1) and , if the initial data holds

and the initial perturbation is and , then the solution of (1) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially

for some constant which satisfies

where

and is the only root of the equation . In addition, is decreasing when and satisfies as and as ;
(2) if , the solution converges to the traveling wave algebraically

Remark 13. In Theorem 12, the assumption

needed in [22] is taken away in this paper. In addition, compared to the stability results obtained in [22], the convergence rates of traveling front solutions with noncritical speeds in this paper are more accurate.

When , then (51) is reduced to

By taking Fourier transform to (105), we get

where

It is clear that

In the same way, by taking the inverse Fourier transform to (106), we have

Then by a similar way of Lemma 6, we can get the following decay rates.

Lemma 14 (decay rates for ). Suppose . Thus, when , the solution of (105) satisfies

Similarly, we can prove the convergence of of to when the time-delay .

Theorem 15 (stability of traveling waves without time-delay). Suppose hold. For the given traveling wave of (1) and with speed , if the initial data holds

and the initial perturbation is in , then the solution of (1) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially

for some constant

where

(2) if , the solution converges to the traveling wave algebraically

Remark 16. From Theorems 12 and 15 we conclude that the time-delay affects not only the initial perturbation, but also the convergence rates of the traveling front solutions with noncritical speeds.

4. The Uniqueness of the Traveling Front Solutions

In this part, the uniqueness of the traveling front solutions will be proved on the premise of the stability.

Theorem 17 (uniqueness of traveling front solutions). Suppose hold. Then, for the same , the traveling front solutions of (1) are unique up to translation.

Proof. For and the same , suppose that and are two different traveling front solutions. Then it follows from Theorem 3 that

and

where are two positive constants and is defined in Theorem 3. Let us move to with some constant . Letting , it is evident that . By choosing

if , we can get

Then, when , there exists such that

which indicates that

Next, by taking the initial data for (1) as

then we get the corresponding solution to (1)

From Theorem 12, for , it holds that

that is, for all if . This completes the proof of the uniqueness of traveling waves up to a constant shift. Similarly, we can obtain the result when the equation is without time-delay.

5. Applications

In this section, we shall present the following results as a direct consequence of Theorems 12 and 15.

5.1. Consider the following Vector-Disease Model (See [30])

where denotes the density of infectious individual at time and site , is the recovery ratio of the infected person, and is the host-vector contact ratio. is the diffusion constant.

Clearly, (125) has two constant equilibria and . The characteristic equation at is

Set

There exists a corresponding number such that . In [30], the authors have justified that is the minimal wave speed of traveling wave solution connecting the two equilibria.

Set and in (1). Obviously, , for , for and

where . Thus we get the following result.

Corollary 18 (stability of traveling waves). When , for the given traveling wave of (125) and , if the initial data holds

and the initial perturbation is and , then the solution of (125) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially

for some constant which satisfies

where

and is the only root of the equation . In addition, is decreasing when and satisfies as . When , then and .
(2) if , the solution converges to the traveling wave algebraically

5.2. Consider the Classical Fisher-KPP Equation (7) with the Initial Data

For (7), it is easy to see that . Obviously, from Theorem 15, we obtain the following result.

Corollary 19 (stability of traveling waves). For the given traveling wave of (7) and with speed , if the initial data holds

and the initial perturbation is in , then the solution of (7) and (134) satisfies the following:
(1) when , the solution converges to the traveling wave exponentially

for some constant

where , ;
(2) when , the solution converges to the traveling wave algebraically

Remark 20. The authors in [18–20] have showed the stability for (7). Our results generalize and develop the former results. The decay rate of the critical waves of (7) is faster than that of [19], but slower than that in [20].

5.3. Consider the following General Population Model, Which Is Evolved from a Mature Population with an Age Structure of a Single Population (See [6])

where are constants and and are Lipschitz continuous functions on any compact interval.

Let . Now we assume that

let , for ..

Clearly, and are two constant equilibria of (139). And the characteristic equation of (139) at is

Set

It has proved that, for any , (139) admits a traveling front solution satisfying and . Then we can show the stability of traveling front solutions of (139).

Corollary 21. Suppose that and hold. When , for the given traveling wave of (139) and , if the initial data holds

and the initial perturbation is and , then the solution of (139) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially

for some constant which satisfies

where

and is the only root of the equation . In addition, is decreasing when and satisfies as . When , then and ;
(2) if , the solution converges to the traveling wave algebraically

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11471197) and the Natural Science Foundation of Shanxi Province (no. 201601D202002).

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