Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay

This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c>c*, where c* is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ(x+ct) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.

They used a sub-super solution method of Wu and Zuo [24] to show the existence of traveling wavefronts, and a squeezing method to get the asymptotic stability of traveling wavefronts. Recently, Guo and Zimmer [7,8] studied the traveling wavefronts of the following equation with nonlocal delay effect ∂u(t, x) ∂t = ∆ 1 u(t, x) + f (u(t, x), (h * u)(t − τ, x)), where (h * u)(t, x) = R h(x − y)u(t, y)dy. Under the monotonicity assumption ∂ 2 f (u, v) ≥ 0 for (u, v) ∈ [0, K] 2 , Guo and Zimmer [7] established the existence, monotonicity, uniqueness and asymptotic behavior of traveling wavefronts of (4). In [8], by using a combination of the weighted energy method and the Green function technique, Guo and Zimmer further proved that all noncritical traveling wavefronts are globally exponentially stable, and critical traveling wavefronts are globally algebraically stable, when the initial perturbations around the wavefront decay to zero exponentially near minus infinity regardless of the magnitude of time delay. It is easy to see that when f (u, v) = −u + g(v) and h(x) = δ(x) (Dirac delta function), (4) reduces to our model (1).
Note that all above results in [1,7,8,18] are established under the assumption that the whole interaction term is quasi-monotone. When the whole interaction term is non-quasi-monotone, there is no results on the traveling waves. In Remark 1.4 of [18], Ma and Zou pointed out that when g(u) is not increasing on [0, K], the traveling wave problem of (1) becomes much harder due to the lack of quasimonotonicity. Inspired by the study on reaction-diffusion equation by Ma [17], in the first part of this paper, we establish the existence of traveling waves of (1) by constructing two auxiliary discrete reaction-diffusion equations with quasimonotonicity. This method can also be seen in [4,10] for establishing the spreading speeds. We should point out that, by this method, we can not get any information on the monotonicity of traveling waves. We leave this problem for the future study.
Our main goal of this paper is to prove the stability of traveling waves of (1). We remark that the methods in [1,8,18] can not be directly applied to our equation H k w (I) is the weighted Sobolev space with the norm given by Let T > 0 be a number and B a Banach space. We denote by C(  with a sufficient large number x 0 1. Now we state the stability result of traveling waves for (1) with a general nonmonotone function g(u). Theorem 1.1 (Stability of traveling waves). Assume that (H1) − (H3) hold. For any given traveling wave φ(x + ct) with c > c * to (1), whatever it is monotone or non-monotone, suppose that exists uniformly with respect to s ∈ [−τ, 0]. Then there exist some constants δ 0 > 0, 0 < µ 2 = µ 2 (τ, g (K)) < 1, and 0 < µ = µ(τ, c, λ, g (K)) < µ 2 , all independent of x, t, u(t, x) and φ(x + ct), such that, when the initial perturbation is small: the unique solution u(t, x) of (1) and (2) exists globally and satisfies and where C unif [−τ, T ] for 0 < T ≤ ∞, is defined by Corollary 1 (Uniqueness of traveling waves). Assume that (H1) − (H3) hold. Then, for any traveling waves φ(x + ct) of (1), whatever they are monotone or nonmonotone, with the same speed c > c * and the same exponential decay at ξ → −∞: they are unique up to translation.
The rest of this paper is organized as follows. In section 2, we study the existence of non-critical traveling waves of (1) with a general nonmonotone function g(u). In section 3, we reformulate the original equation to the perturbed equation around the given non-critical traveling wave, and then give the corresponding stability theorem for the new equation. In section 4, we take the weighted energy method to establish the desired a priori estimates, which play an important role in the proof of stability. Based on the stability theorem, in section 5, we prove the uniqueness of those monotone or nonmonotone traveling waves. In section 6, we give some applications.
2. Traveling waves. In this section, we study the existence of non-critical traveling waves of (1) without monotonicity on g(u). A traveling wave for (1) connecting with two steady states 0 and K at far fields is a special solution in the form of u(t, x) = φ(x + ct) ≥ 0. Substituting φ(x + ct) into (1) and letting ξ = x + ct, we obtain the following wave profile equation with the boundary conditions and c is the wave speed. It is clear that the characteristic function for (8) with respect to the trivial equilibrium 0 can be represented by One can easily show that the following result holds. Furthermore, if c > c * , then P(c, λ) = 0 has two distinct positive real roots λ 1 (c) and λ 2 (c) with λ 1 (c) < λ * < λ 2 (c), and P(c, λ) > 0 for λ ∈ (λ 1 (c), λ 2 (c)).
When g(u) is increasing on [0, K], the existence of traveling wavefront have been established in [18] by using sub-super solutions and monotone iteration technique.
From Lemma 2.2, we can obtain the following existence result on traveling wavefronts for (9) and (10).
for all ξ ∈ R. Then we further obtain is well defined. Furthermore, it is easily seen that a fixed point of F is a solution of (8).
3. Reformulation of the problem. This section is devoted to the proof of stability of those monotone or non-monotone non-critical traveling waves of (1). Let φ(x + ct) = φ(ξ) be a given traveling wave with speed c > c * , and

STABILITY OF NON-MONOTONE NON-CRITICAL TRAVELING WAVES 589
Then, from (1) and (8), we can see that v(t, ξ) satisfies where Let T > 0. We define the solution space for (19) as follows Particularly, when T = ∞, we denote the solution space by X(−τ, ∞) and the norm of the solution space by M v (∞). Now we state the stability result for the perturbed equation (19) which automatically implies Theorem 1.1.
By using the continuity extension method [15,16], the global existence of v(t, ξ) directly follows from the local existence result and the a priori estimate given below.
. By Taylor's formula and (H3), we obtain where η is some function between φ and φ + v (n) . It is easy to see that the solution of (20) can be written in the integral form Combining (21) and (22) Furthermore, by Gronwall's inequality, we get Notice that v (n) (t, ξ) ∈ C unif [−τ, t 0 ], namely, lim . We rewrite the solution of (20) as It is clear that
The proof for the a priori estimates of the solution in the designed solution space X(−τ, T ) is technical and plays a crucial role in this paper. We leave this for the next section.
Proof of Theorem 3.1. Let δ 2 , C 0 and µ be the positive constants given in Proposition 2 which are independent of T and v, andc 0 be a constant given in Proposition 1. Now let us choose By Proposition 1, there exists t 0 = t 0 (δ 1 ) > 0 such that v(t, ξ) ∈ X(−τ, t 0 ). From the selection of δ 0 and δ 1 , we can further confirm M v (t 0 ) ≤ δ 2 . Then by Proposition 2, we can obtain the exponential decay estimate (32) for t ∈ [0, t 0 ]. Next, let us consider (19) with the new initial data v(s, ξ) for s ∈ [t 0 − τ, t 0 ]. Again, by Proposition 1, we can prove that the solution to the new Cauchy problem (19) exists for time t in [t 0 , 2t 0 ]; namely, we extend the time interval of the solution to [−τ, 2t 0 ], or say, v(t, ξ) ∈ X(−τ, 2t 0 ). Then, by using Proposition 2, we can establish the exponential decay estimate (32) for t ∈ [0, 2t 0 ]. Repeating this procedure, we can prove global existence of the solution v(t, ξ) ∈ X(−τ, ∞) with the exponential decay estimate (32) for t ∈ [0, ∞]. For details, we refer the reader to [16]. The proof is complete.

4.
A priori estimates. In this section, we shall establish the a priori estimates. The adopted approach is the weighted energy method but with a new development.
We first get the energy estimates for v(t, ξ) in the weighted Sobolev space H 1 w (R). where and and µ and η both are arbitrarily give positive constants at this moment, but will be specified later.
Then we take the limit ε → 0 to obtain the corresponding energy estimate for the original solution v(t, ξ). For the sake of simplicity, below we give the formal calculation using v(t, ξ) directly to establish the desired energy estimates. Multiplying (19) by e 2µt w(ξ)v(t, ξ), where µ > 0 is a constant and will be specified later in Lemma 4.3, we have By (36), we get Integrating (37) over R × [0, t] with respect to ξ and t yields By a similar argument as in Proposition 1, we further have By the Cauchy-Schwarz inequality, one has for any η > 0. Furthermore, by change of variables, we obtain Substituting (39) into (38) yields Namely, where B η,µ,w (ξ) is given in (34).
On the other hand, by the definition of M v (T ), we have By the Taylor's formula, we get STABILITY OF NON-MONOTONE NON-CRITICAL TRAVELING WAVES   597 where C > 0 is independent of v. Then we can estimate the nonlinear term as e 2µs w(ξ)|v 0 (s, ξ)| 2 dξ ds.
Proof. By (42), one can estimate B η,µ,w (ξ) as defined in (34)  Next, we establish the estimate for the one order derivative v ξ (t, ξ) of the solution v(t, ξ).
Proof. Differentiating (19) with respect to ξ and multiplying it by e 2µt w(ξ)v ξ (t, ξ), then integrating the resultant equation with respect to ξ and t over R × [0, t] and applying Lemma 4.3, we can similarly prove (45). Thus, we omit the details.
Remark 1. As discussed in the proof of Lemma 4.1, in order to establish the energy estimate (45), we need a good enough regularity for the solution v(t, ξ). Actually, the same mollification procedure is also needed in the proof of Lemma 4.4. Here, we omit the details.
Finally, combining Lemmas 4.3 and 4.4, we obtain the following a priori estimates.