Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\R^d}K(y)g(v(t-h,x-y))dy, x \in \R^d,\ t>0;$ where $h>0$ and $d\in\Z_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d=1$ which includes the global stability on Sobolev spaces. Here $g$ is not assumed to be monotone and the kernel $K$ is not assumed to be symmetric, therefore non-monotone semi-wavefronts and {\it backward traveling fronts} appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.


Introduction
We study the following non-local equation with delaẏ v(t, x) = ∆v(t, x) − v(t, x) + In general, the study of delayed case mainly presents two troubles. The first one is concerned with the asymptotic behavior of semi-wavefronts in the positive equilibrium κ since the associated characteristic equations have infinity solutions and semi-wavefronts could oscillate around κ. Indeed, non-monotone wavefronts to (1) have been observed [36,39,13,43]. Otherwise, the second trouble is that the associated semi-flow to (1) is not monotone in general. This lack complicates the construction of sub and super-solution, an approach widely used when h = 0 or g is monotone to prove the existence and stability of wavefronts. The spectral technique has been used in order to obtain the local stability [11,19,28,29], however, the maximum principle arguments to reaction-diffusion equations frequently imply the global stability of wavefronts [2,38,26,33,18]. Nevertheless, our approach is a combination of maximum principle arguments and Fourier analysis for linear delay PDE's.
For local equations (with d = 1), when g is sub tangential and possibly non-monotone the local exponential stability of wavefronts, in suitable Sobolev spaces, was given by Lin et al [24] (for non-critical wavefronts) and Chern et al [6] (for critical wavefronts) under the condition |g ′ (κ)| < 1 for any delay or g ′ (κ) < −1 for small delay. Then, in [24] the algebraic stability of semi-wavefronts with speed c ≥ c(|g| Lip ), some speed c(|g| Lip ) (see Definition 2.5 below), on any domain of the form (−∞, N], N ∈ R, was proved in [32,Theorem 3] without assumptions on neither subtangetiality of g nor size of derivate on equilibrium κ. In particular, when |g| Lip = g ′ (0) semi-wavefronts (including the critical and asymptotically periodic semi-wavefronts) are stable on any domain (−∞, N]. This limitation on the stability domain is by the use of an unbounded weight so that the control of the stability of semiwavefronts on its whole domain yields to the stability with a bounded weight. For the local stability in [32,Corollary 17] was showed that the size of local perturbations depends on the size of neighborhoods of κ where g is contractive application and one of these neighborhoods of κ is attractor and therefore the global exponential stability of non-critical semi-wavefronts was also established in [32,Corollary 11], which includes non-monotone wavefronts for typical models such as local Nicholson's blowflies model (when p/δ ∈ [1, e 2 ]) and Mackey-Glass' model. However, the stability of critical semi-wavefronts was not addressed in [32] so that we study the global stability of critical wavefronts in this paper.
In respect to the non-local equations, when d = 1 the stability of wavefronts has also been studied for bistable nonlinearity without delay (see, e.g., [5]) and with delay (see,e.g., [40] and [25]). In the monostable case (1) with delay, the global stability of the monotone wavefronts with monotone g was satisfactorily answered by Mei et al in [26] when K is a heat kernel. Similar results, for more general equation, were established by Lv and Wang [21]. Also, a close model to (1) is a paper of Wang et al [41] where the authors proved the global stability of non-critical (under minimal conditions on the initial data) when g is monotone and K is an even kernel. For d ≥ 1, we should mention a very interesting work for dispersal equations presented by Huang et al [18] where the global stability of monotone planar wavefronts was stated and the study of the convergence rate was dealt; here K is a multidimensional heat kernel. So that, as much as we know the study of stability of semi-wavefronts for the non-local case assumes the monotonicity of g and the symmetry of K. Thus, our aim is to prove the global stability of wavefronts which could be backward wavefronts or oscillatory wavefronts. In particular, our global stability result for asymmetric kernel implies a change of behavior in the problem of speeds selection for the equation (1), i.e., to determinate the asymptotic speed propagation of solutions generated by an initial data by only knowing the asymptotic behavior of the initial data at the trivial equilibrium (see Remark 2.9).
In respect to the convergence rate of solutions to critical semi-wavefronts our result of local stability is comparable to Gallay work [11] for local equations without delay. More precisely, the disturbances space in [11] is a subspace of our disturbance space in the sense that the weights defer by a quadratic factor. Although, in our space the convergence is O(t −1/2 ) while in the subspace considered in [11] is faster than O(t −3/2 ). Also, the convergence rate in our global stability result extends the pioneering result of Mei et al [26] in Sobolev spaces for (1) (see Corollary 2.10 below) without requiring the convergence of the initial datum to κ. This paper is matched with a recent work of Benguria and Solar [4] where it is showed that the algebraic convergence rate for critical wavefronts obtained in this paper is optimal in the underline weigthed space and it also has a closed relation with convergence rate obtained in [18].
We organize this paper in the following way. In the Section 2 we present and discuss the main results, in the Section 3 we state an existence and regularity result for the Cauchy problem, in the Section 4 we prove the stability results for d ≥ 1 (stability on semi-intervals and local stability) and finally, in Section 5 we prove the global stability result for d = 1.

Main Results and Discussion
2.1. Global stability with exponential weight on R d . Now, in order to study the stability of semi-wavefronts with speed c in the direction ν ∈ S d−1 we make the change of variables z := x + ctν and u(t, z) := v(t, z − ctν), so that we have the following equation for u for which the planar semi-wavefronts v(t, x) = φ c (ν · x + ct) with speed c, φ c : R → R + , are stationary solutions u(t, z) = φ c (ν · z) the following equation |u(s)| L r,p λ } When r = 0 the letter W is replaced by L. Analogously, we define weighted Holder spaces C r,p λ and C r,p h,λ . Finally, for some function u : [a − h, b] → X, some Banach space X and a, b ∈ R, we define Now, we state our first result on the stability of solutions to (3).
then u t (·, ·) and ψ t (·, ·) uniquely exist in L ∞ h,λ ∩ C 0,α h,λ for all t ≥ −h. Moreover, if r(t, z) satisfies (5) with the initial datum ξ λ r 0 we obtain and γ λ is the unique real solution of the following equation In particular, if φ c is a stationary solution of (3) and q λ ≤ −p λ , i.e. γ λ ≤ 0, then φ c is Remark 2.2. When the initial datum is taken in C 0,0 h,λ = BUC(R d ) it is possible to prove the existence of a mild solution of (1) on (0, +∞) × R, see Remark 3.3. Moreover, it also is possible to prove that for t > h that solution is a classic solution of (1), see Proposition 3.6 and compare with [41,Theorem 3.3].
Corollary 2.4 (Uniqueness of semi-wavefronts). Assume the condition (L). If φ c andφ c are stationary solutions of (3) Naturally, because of (7) the perturbation is maintained in the space C([−h, 0], L 1 (R) ∩ L ∞ (R)) with weight ξ λ for all t > h(d + 1)/2. This fact is true for all t ≥ −h as it is showed in Proposition 3.6. For g ∈ L ∞,k (R) we give a result on the persistence of the derivates of perturbations in Section 3 which shows that for the derivates of order k the persistence is obtained for t > h(k − 1).
We note that the Cauchy problem to (3) is well posed for non negative initial data since an application of maximum principle on unbounded domains (see, [27,Theorem 10,Chapter 3]) implies that u(t, ·) is positive for t ∈ [0, h] and repeating this argument to intervals [h, 2h], [2h, 3h], ... we can conclude that u(t, ·) is positive for all t > 0.

2.2.
Existence of d-dimensional planar semi-wavefronts. The results on the existence of wavefronts of (4) for monotone g in an abstract setting are well known [42,22,35]. For non-monotone g, the existence of non monotone wavefronts to (4) has been studied when d = 1 [17,36,39,13,43]. For d > 1 and K satisfying the following condition (K) There exist non-negative functions K i ∈ L 1 (R), for i = 1, ..., d, such that Also, the function λ ∈ R → R K i (y)e −λy dy, is defined on some maximal open interval (a i , b i ) ∋ 0, the existence result for planar semi-wavefronts given in [13,Theorem 18] can be applied to (4). For instance, in [18] the authors take K i equal to a heat kernel for all i = 1, ..., d.
More precisely, associated to equation (4), for each c ∈ R, we have the characteristic Without restriction of (K) we can take R K i (s)ds = 1 for i = 1, ..., d. Next, if we fix a canonic vector e, let us say e = e 1 , then E c := E 1 c defined on the maximal open interval (a 1 , a 2 ) =: (a, b) ⊂ R is the characteristic function associated to trivial equilibrium for wave's equation (4) and therefore according to [13,Lemma 22] we can make the following definition Also, if c ≤ c − * then the zeros of E c are negative while if c ≥ c + * then the zeros of E c are positive. Therefore, because of E c (0) > 0 and the continuity and monotony of E c on the parameter c the function E c is not positive in the compact interval defined by zeros of E c . Now, in order to establish the next results we make the following mono-stability condition.
Under conditions (M) and (K) the existence of semi-wavefronts was established, e.g., in [13, Theorem 18] and we present it as follow.
Proposition 2.6. Suppose that g satisfies (M) and K satisfies (K). Then for each c ∈ C the equation (1) Also, if for some ζ 2 = sup s≥0 g(s) the equilibrium κ is a global attractor of the map g : (0, ζ 2 ] → (0, ζ 2 ], then each semi-wavefront is in fact a wavefront.
In the particular case when g is monotone, Proposition 2.6 says that semi-wavefronts for non-local equation (1) are wavefronts, indeed these are monotone wavefronts (see Remark 5.5 ). The problem in determining the condition for which κ is a global attractor for g : (0, ζ 2 ] → (0, ζ 2 ] was dealt in [44] where the following condition characterizes this globalness property.
(G) The application g 2 has a unique fix point κ on (0, ζ 2 ]. In this sense, under condition (G) and an additional hypothesis on K (which can be dropped by Proposition 2.6) the authors in [43] have stated the existence of minimal speed for the existence of wavefronts.

2.3.
Local stability of d-dimensional planar waves. Following notation of Subsection 2.1 we denote E c (λ) = q λ + p λ . Also, for some λ ∈ R we define the bounded weight function and define the space for certain C ǫ ∈ (0, 1/2], then the following assertions are true It is instructive to compare Theorem 2.7 with a work of Gallay [11] about the local stability of critical wavefronts to a local equation with h = 0. Note that in [11] the perturbation is additionally weighted with quadratic function in the trivial equilibrium and the exponential convergence to the positive equilibrium is assumed. Although, in this subspace considered by Gallay the rate the convergence is as O(t −3/2 ). Otherwise, note that for non-critical semiwavefronts the convergence rate depends on the weighted space where the perturbation is taken attaining an algebraic convergence rate when the perturbation is in C([−h, 0], L 1 (R)) with weight η λ j (c) , j = 1, 2, and an exponential convergence rate if λ ∈ (λ 1 (c), λ 2 (c)).

2.4.
Global stability of wavefronts on the line. In this section we take d = 1 and give a global result in the sense that the wavefronts are attractors for the following class of initial data (3) is a bounded function and there are σ > 0 and z 0 ∈ R such that : We note that in Theorem 2.7 it is necessary ǫ < κ and φ c (z) ≥ κ − ǫ/2 for z ≥ z ǫ , therefore an initial datum satisfying the condition (11) meets the condition (IC) with σ = κ − ǫ and z 0 = z ǫ .
This result includes the classic Fisher-KPP model when h = 0 and K is the Dirac function. We also have the following result for non-local Nicholson's model (2). If φ c is a wavefront with speed c ∈ C to (2) then φ c is either globally algebraically stable in B 1 h,λ 1 or globally exponentially stable in B 1 h,λ whenever E c (λ) < 0, in the sense of Theorem 2.8. For a local version of (2) in [12,Theorem 2.3] it was demonstrated that for p/δ ∈ [e, e 2 ) this equation has non-monotone wavefronts with speed arbitrarily large. Under this restriction on the parameters p and δ, Solar and Trofimchuk have demonstrated the global stability of non-critical wavefronts for the local Nicholson equation [33,Corollary 3]. Thus, the global stability of critical wavefronts for local equations in Corollary 2.11 is a complement to the result obtained in [33].

A Regularity Result
We start giving a result on the persistence of disturbances in the underlying space for the following equatioṅ h,λ ′ some 1 ≤ p ≤ ∞ then we have the following estimate for the associate solution u(t, z) to (16) for some θ = θ(λ ′ ) > 1.

Proposition 3.2.
Suppose that g is globally Lipschitz continuous and K ∈ L 1 λ ′ ∩ L 1 λ for some λ ′ , λ ∈ R. If the initial datum u 0 ∈ L ∞ h,λ ′ ∩ C 0,α h,λ then there exist a unique solution u(t, z) to the nonlinear equation (3) and the solution u(t, z) satisfies the estimation (17) and
Proof. If k = 1 by Lemma 3.4 we have u z i (t, z) ∈ L p , for each t ∈ (0, h] uniformly (in norm) on compacts. Moreover, by Remark 3.5, u z i (t, ·) ∈ L p (R), for each t ∈ (0, +∞), uniformly (in norm) on compacts. In particular, if T > h then u z i (t + T, ·) ∈ L p h , therefore we analogously conclude u z j z i (t, ·) ∈ L p , for each t ∈ (h, +∞), uniformly (in norm) on compacts. The same argument is applied for k = 3, 4.... in order to obtain (24).
Lemma 4.1. The function l λ meets the following inequalities Remark 4.2. In the local case (whenk is formally a constant q) we have e l(ζ) ∼ −q/ζ 2 (see [32,Lemma 13]) but in the non local case, because of Riemann-Lebesgue Lemma, the estimations for l(·) can be improved.
From Lemma [32, Lemma 12] we have that β(ζ) ≤ 0 if and only if: Now, by using log(1 + x) ≤ x, for all x ≥ 0, then in order to obtain (27) it is enough to have This proves (26).
Proof of Theorem 2.1 Note that by Proposition 3.2, u(t, ·) and ψ(t, ·) exist uniquely in L ∞ h,λ . Then, by making the following change of variableũ(t, z) = u(t, z)e −λ·z we havė

5.
Proof of Theorem 2.8 5.1. Monotone case. We begin this section with some results which generalize those founded in [33] and [32]. In this section, g : R + → R + is a monotone function which is extended linearly and C 1 on (−∞, 0].
where the nonlinear operator N is defined by The definition of a sub-solution u − is similar, with the inequalities reversed in (36).
Hence, the usual maximum principle holds for each Π r , r ≥ r 0 , so that we can appeal to the proof of the Phragmèn-Lindelöf principle from [27] (see Theorem 10 in Chapter 3 of this book), in order to conclude that δ ± (t, z) ≤ 0 for all t ∈ [0, h], z ∈ R.

5.2.
Attractivity of an optimal neighborhood of κ.
for some N > 0.
If g 1 or g 2 is a non-decreasing function, then implies v 1 (t, z) ≤ v 2 (t, z) for all (t, z) ∈ R + × R.
So by (64) and Phragmèn-Lindelöf principle we conclude that and by (64) Therefore using again (64) and (67) Finally, by (62) and (68) Otherwise, for c ≤ c − * if we use the inequality (57) and the same function β(t) then the situation is completely analogous and therefore (58) can be obtained.