Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed \begin{document}$c>c_*$\end{document} , where \begin{document}$c_*>0$\end{document} is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ( \begin{document}$c = c_*$\end{document} ), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.


1.
Introduction. The object of this paper is to consider a nonlocal time-delayed reaction-diffusion equation with the following initial data u(s, x) = u 0 (s, x), (s, x) ∈ [−r, 0] × R. ( Here u(t, x) denotes the total mature population of the species at time t and position x, D > 0 is the spatial diffusion rate for the mature population, r > 0 is the maturation delay, the time required for a newborn to become matured. Here α > 0 is the total amount of the immature species, and f α (y) is the heat kernel in the form of The nonlinear functions d(u) and b(u) denote the death and birth rates of the mature population respectively, and satisfy the following hypotheses: (H 1 ) There are two constant equilibria saying u ± : u − = 0 is unstable and u + > 0 is stable. That is d(u ± ) = b(u ± ), and d (0) < b (0), d (u + ) > b (u + ); (H 2 ) b(u) ∈ C 2 [0, ∞), b(u) ≥ 0, and |b (u)| ≤ b (0) for u ∈ [0, ∞); As far as we know, the non-monotonicity of b(u) causes the delayed reactiondiffusion equation (1) to be non-monotone and loss its comparison principle. The corresponding traveling waves solutions may also be oscillating when the time delay is big or the wave speed is big.
In this case, the nonlocal equation (1) is reduced to a local time-delayed reactiondiffusion equation for α = 0: ∂u(t, x) ∂t − D ∂ 2 u(t, x) ∂x 2 + d(u(t, x)) = b(u(t − r, x − y)). A traveling wave of (1) connecting the constant states u ± is a special solution of the form of φ(x + ct) , where c > 0 is the wave speed. Thus, it satisfies In this paper, we will show that these traveling waves, including non-monotone noncritical/critical waves, are asymptotically stable as t −→ ∞.
The non-local reaction-diffusion equations with time delay are the important and interesting models from both physics and biology. For the non-critical traveling wave with wave speed c > c * , Mei et al. [19] obtained the exponential stability by using combination of the weighted energy method and the companison principle for the monotone equation. Later on, the global stability of critical wavefronts with optimal convergent rates was proved by Mei-Ou-Zhao [20] by using the Fourier transform and Green's function method plus energy estimate. For the local equation, by using the L 2 -weighted energy and Hanalay's inequality, Lin-Lin-Lin-Mei [14] first obtained the exponential stability for the non-critical oscillating waves. Then, the stability of critical traveling waves was obtained by Chern-Mei-Yang-Zhang [2] by using the anti-weighted technique and energy estimates. For nonlocal diffusion problem with time delay, Huang-Mei-Wang [10] got the global stability and the optimal rates for the planar wavefronts by Fourier transform and the weighted energy method. Recently, using the anti-weighted method, the stability of oscillating traveling wave for time-delayed nonlocal dispersion equations was obtained by Huang-Mei-Zhang-Zhang [11]. On the other hand, for the initial boundary values problem, Jiang-Zhang [12] obtain the global stability for the monotone equation by using the weighted energy method and the squeeze theorem, and the local stability for the non-monotone equation by using the weighted energy method for the noncritical wave.
For the precious works, Mei-Lin-Lin-So [19], Huang-Mei-Wang [10] and Mei-Ou-Zhao [20] showed the stability for traveling waves but it sufficiently depends on the advantage of the monotonicity of both the equation and the traveling waves. Notice that, in this paper, the equation loses monotonicity and the traveling wave may be oscillating when the time-delay r is sufficiently big. So, the above approaches, including Fourier transform method, the monotone technique and L 1 -weighted energy method seem to fail in getting the stability of the traveling waves for (1). On the other hand, since the nonlocal birth term exists, the regular L 2 -weighted energy method [14] can not be applied to deal with the stability of these slower waves.
In this paper, the equation (1) is nonlocal and non-monotone, and the traveling waves may be oscillating when the delay r is large. Inspired by the study on the p-system with viscosity by Matsumura-Mei [17] and the study on the stability of the critical traveling waves for time-delayed reaction-diffusion equations with local non-monotone nonlinear term [2], where they give a suitable transform function (i.e. the anti-weight) to change the equation to a new equation, we realize that we can overcome the difficulty caused by the integral terms in the L 2 weighted energy estimates. Similar to [14], we work out the trouble caused by these oscillations by the nonlinear Halanay's inequality.
The rest of the paper is organized as follows. In Section 2, we give our main stability results after introducing some necessary notations. When c > c * , the solution will be expected to exponentially converge to its corresponding traveling wave if the initial perturbation around the wave is small enough in a certain weighted Sobolev space. However, when c = c * , we can only obtain the time asymptotic stability for the critical wave. In Section 3, we reformulate the original equation to the perturbed equation around the given traveling wave and give the corresponding stability theorem for the new equation. In Section 4, we use the anti-weighted technique to establish the desired a priori estimates and use the nonlinear Halanary inequality to treat the case when the traveling wave near u + . This plays a crucial role in the proof of stability.
2. Preliminaries and main results. At first, we state some notations used throughout this paper. C denotes a generic constant, while C i > 0 (i = 0, 1, 2...) represents a specific constant. Let L 2 (R) be the space of the square integrable functions defined on R, and H k (R)(k ≥ 0) is the Sobolev space of the L 2 -functions f (x) defined on the R whose derivatives d i f dx i (i = 1, ..., k) also belong to L 2 (R). Let T be a positive number and B a Banach space. We denote by C([0, T ]; B) to be the space of the B-valued L 2 -function on [0, T ].
for some positive constants λ + and λ * + determined by For the exciting traveling waves, from [7], [14], [28] and [29], we know that the traveling waves may be non-monotone and oscillatory around u + , when the timedelay r is suitably big. Furthermore, when d (u + ) < |b (u + )|, there exists a positive constant r, such that, if the time-delay r satisfies r > r, there will be no traveling waves [3]. As far as we know, when traveling waves lose monotonicity, the perturbation equation around the traveling wave will be not monotone. Furthermore, the monotone technique can be no longer applied for the stability of the traveling waves.

Main results.
As follows, we give the main results of this paper. Let φ(x+ct) be a given traveling wave with c ≥ c * , even if the traveling wave is monotone or slowly oscillatory around u + . Here c * = c * (r) and λ * = λ * (r) satisfy (6). We define two weight functions by where λ > 0, λ 1 and λ 2 , stated in (8), are the eigenvalues of the characteristic equation (5), and λ * stated in (6). Now we state the asymptotic behavior of non-critical and critical traveling waves.
For given traveling wave φ(x+ct) with c > c * to (1), no matter the wave is monotone or oscillatory. Suppose that When the initial perturbation is small: for some positive number δ 1 , then the solution u(t, x) of (1)-(2) is unique, and exists globally in time, and satisfies for some constant Theorem 2.2 (Stability of critical traveling waves). Under the assumptions of Theorem 2.1, for the critical traveling wave φ(x + c * t), suppose that the initial perturbation and is small enough, namely, there exists a constant δ 2 > 0, such that Then the solution u(t, x) of (1)- (2) is unique, and exists globally in time, and satisfies Remark 1. In Theorem 2.2, the critical traveling waves φ(x + c * t), no matter they are monotone or oscillatory, are proved to be time-asymptotically stable when the initial perturbations are small enough. However, due to some technical restrictions, the convergence rate can not be obtained in this paper.
3. Reformulation of the problem. This section is devoted to the proof of the Theorem 2.1 and Theorem 2.2 for the stability of those non-critical and critical traveling waves of (1). Let φ(x + ct) = φ(ξ) be a given traveling wave with the speed c > c * , and u(t, x) be the solution of (1) with a small initial perturbation around the wave φ(x + cs) Then, from (1)- (2), v(t, ξ) satisfies where Let 0 ≤ T ≤ ∞, we define the solution space for (20) as follows equipped with the norm For the critical wave, similarly as the non-critical wave, we denote The solution space for (26) is defined as follows. Let 0 ≤ T ≤ ∞, equipped with the norm Next, we state the corresponding stability result for the initial value problem (20) and (26).
Notice that, Theorem 2.1 is equivalent to Theorem 3.1, and Theorem 2.2 is equivalent to Theorem 3.2.
4. Proof of the main results. In this section, we are going to show the main results. The adopted method is the so-called transformed energy method (i.e. antiweighted method) combining with the noninear Halanay's inequality.
4.1. Non-critical traveling wave. At first, we give the following transformation. (20), then we derive the following equation forṽ(t, ξ) Next, we have the a priori estimates by several lemmas.
Proof. Multiplying (31) by e 2µtṽ , where µ > 0 will be selected later, we get Integrating (33) over R ×[0, t] with respect to ξ and t, using the property of Sobolev space H 2 (R), we further have Again, by using Cauchy-Schwarz inequality and by change of variables, we have Substituting (35) into (34), we get Next, we estimate the nonlinear terms.
Combining Lemma 4.1 and Lemma 4.2, we get the following estimate. Let v(t, ξ) ∈ X(−r, T ), then there exists a constant µ > 0, such that it holds that provided N (T ) 1.
Next we derive the estimates for the higher order derivatives of the solution.
Lemma 4.4. It holds that Proof. Differentiating (31) with respect to ξ and multiplying it by ∂ṽ ∂t , then integrating the resultant equation with respect to ξ and t over R×[0, t], we can similarly prove (43) provided N (T ) 1. The detail is omitted.
Thus, combining (42) and (43), we will establish the following energy estimate.
Lemma 4.5. It holds that By using the Sobolev inequality, we can get the following lemma.
Lemma 4.6. Let ξ 1 be a big enough positive constant, then it holds that provided ξ ≤ ξ 1 .

4.2.
Critical traveling wave. For the critical waves, since the inequality (39) is no longer satisfied , the decay rate can not be obtained by the same method for the non-critical waves. Inspired by the reference [2], we can only get the time-asymptotic result by the anti-weighted energy method. Taking the following transformation (or say, anti-weight) we get the following equations forṼ (t, ξ) Now we are going to establish the estimates of the solution V ∈ Y (−r, T ) by several lemmas. Lemma 4.9. It holds that provided N (T ) 1.
In order to prove Lemma 4.9, we need the following lemmas.
Lemma 4.10. It holds that Proof. Multiplying Eq. (47) byṼ and integrating it with respect to ξ and t over R × [0, t], we have By using Cauchy-Schwarz inequality and by change of variables, we get Substituting (51) into (50), we get (49).
Now we are going to give the estimate of A(ξ).
Next, we will establish the estimates of the nonlinear terms.
Lemma 4.12. It holds that Therefore, Thus, Similarly as the proof of (56), we can get (57).
From Lemma 4.10 to Lemma 4.12, we can easily show Lemma 4.9. Next, we establish the estimates of the higher order derivatives of the solution.
Lemma 4.13. It holds that Proof. Differentiating (47) with respect to ξ, we get Multiplying (62) byṼ ξ and integrating it with respect to ξ and t over R × [0, t], as showed in Lemma 4.10, we have Next, we give the estimates of I 1 , I 2 and I 3 .
Rewriting Q 1 (V ), and we get