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.
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  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  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  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  developed and improved the wave stability results showed in . By using the above methods, Wu et al. in  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  gave the local stability of the traveling waves including the critical waves (no convergence rate). Immediately, Moet  proved that the critical waves of the KPP equation were algebraically stable by using the Green function method. Later, Kirchgässner  and Gallay  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  and Wang  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  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.  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 , 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  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 , 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.
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 ). 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 ). 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
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 .
Now we need the following three steps.
Firstly, we will prove the convergence of to
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 .
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 thatand where .
Moreover, we need to prove the decay rate of for .
Lemma 8. There holds thatwith some constant which satisfies andwhere , , 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.
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