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

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.

From ( 2 ), it is easy to see that  − = 0,  + =  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 reactiondiffusion 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][12][13][14][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 reactionadvection-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 timedelayed 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 reactiondiffusion 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  1 -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 reactiondiffusion 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 timedelay 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.

Preliminaries
In this section, we introduce some notations as follows.We assume that  > 0 represents a general constant and   > 0 denotes a concrete constant.Set  to be an interval, ordinarily  = R.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 B represents a Banach space.Also we let ([0, ]; B) be the space of the B-valued continuous functions on[0, ].
Similarly, we can define the corresponding spaces of Bvalued functions on [0, ∞).
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  > 0 Thus where and   2   is the delayed exponential function defined by and   2   is the fundamental solution as Lemma 2 (see [23]).Set  1 ≥ 0 and  2 ≥ 0. Thus the solution () of ( 13) satisfies where and the fundamental solution   2   with  2  > 0 of ( 17) satisfies for arbitrary constant  > 0.
and the solution of (13) satisfies where  1 is uniquely determined by Moreover, clearly, conditions ( 1 ) − ( 3 ) guarantees the existence of the traveling front solutions of (1) which was showed in [25,26,29].So we have the following result.

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].
(49) Take then V(, ) satisfies the following equation: where When  > 0, 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 (60) Now we will prove the asymptotic behavior of V(, ).
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 V(, ) ≤ Ṽ(, ) = V(, ) (− 0 ) ≤ V(, ) for  ≤  0 , we can directly get the following result for V(, ).

Lemma 7.
There holds that and where Moreover, we need to prove the decay rate of V(, ) for  ≥  0 .Lemma 8.There holds that with some constant ] which satisfies and where Proof.Now we consider the following equations, for  >  * : 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.

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.

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

Consider the following
Vector-Disease Model (See [30]) where  >  > 0, (, ) 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.
for some constant ] which satisfies where For (7), it is easy to see that  − = 0,  + = 1.Obviously, from Theorem 15, we obtain the following result.
The decay rate of the critical waves of ( 7) is faster than that of [19], but slower than that in [20].

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