Stability of solutions to functional KPP-Fisher equations

We study the stability properties of semi-wavefronts of the KPP-Fisher equation with inﬁnite delay ∂∂t u ( t, x ) = ∂∂x 2 u ( t, x ) + R + ∞ 0 u ( t − s, x ) dµ 1 ( s ) (cid:16) 1 − R + ∞ 0 u ( t − s, x ) dµ 2 ( s ) (cid:17) , t > 0 , x ∈ R , where µ 1 and µ 2 are Borel measures. We make an interesting remark about the non convergence in form when the delay is ﬁnite, unlike the classic convergence result to KPP-Fisher equation without delay. We also present a result about the stability of semi-wavefronts to the Neutral KPP-Fisher.


Introduction
The main goal of this paper is to present some stability results for two types versions of the KPP-Fisher equation with delay, i.e., the KPP-Fisher with infinite delay for Q ∈ L 1 (R ≥0 , R ≥0 ) satisfying +∞ 0 Q(s)ds = 1 and the Neutral KPP-Fisher where h > 0 and 0 ≤ b < 1. Both (1) and (2) are a generalization of the classic KPP-Fisher with local delay, also called Hutchinson Diffusive Equation, There is many open problems to the equation (3), mainly related to wavefronts. A wavefront for (3) is a bounded solution in the form u(t, x) = φ c (x + ct), for some speed c > 0 and some profile φ c : R → R + satisfying φ c (−∞) = 0 and φ c (+∞) = 1; if the last condition is replaced by lim sup x→+∞ φ c (x) > 0 then φ c is called a semi-wavefront; a semi-wavefront which is not a wavefront is called a proper semi-wavefront. It is known that there is a minimal speed c * = 2 to the existence of semi-wavefronts to (3) (see [8] and [3]). A big open problem is to establish whether a semi-wavefront is actually a wavefront. In this respect, Hasik and Trofimchuk [7] showed that if h ≤ 3/2 then each semi-wavefront is a wavefront, while if h ≥ 1.861 . . . then each semi-wavefront is a proper semi-wavefront; indeed proper semiwavefronts are asymptotically periodic [3].
On the other hand, the stability techniques to the evolution equation (3) have allowed to obtain some properties of wavefronts [1]. For instance, it has been proved that a semi-wavefront φ c is unique (up to translation) for all c ≥ 2 √ 2 (to a complete solution of uniqueness problem see [20]), so that for given h ≥ 0 and c ≥ 2 √ 2 there is no simultaneously a proper semi-wavefront and a wavefront with speed c to (3). The stability results in [1] and [21] use a suitable exponential weight. More precisely, if c ≥ 2 √ 2 then for certain λ * ∈ (λ 1 , λ 2 ), where λ 1 = (c − √ c 2 − 4)/2 and λ 2 = (c + √ c 2 − 4)/2 , the condition implies the existence of γ < 0 such that It is worth noting that if the used weight in (5) is a bounded wight ρ 0 , i.e., then u 0 (s, x) → 1 as x → +∞, uniformly to s ∈ [−h, 0], implies φ c is a wavefront. More precisely, Proposition 1.1 Let u 0 be a bounded initial datum to equation (3) satisfying, locally, the Hölder condition in x. If u 0 (s, x) → 1 as x → +∞, uniformly on s ∈ [−h, 0], then for each T > 0 Proof: By [5], for t ∈ [0, h] there is a unique solution u(t, x) to (3) which is uniformly bounded on [0, h] × R which is smooth function and where E(t, x) = e −x 2 /4t /2 √ πt. By passing to limit as x → +∞ in (8) we conclude u(t, x) → 1 as x → +∞, uniformly to t ∈ [0, h], the result is obtained by repeating the argument to intervals [h, 2h], [2h, 3h], . . . Therefore, for an initial datum as in Proposition 1.1 the inequality (6) yields to φ c (+∞) = 1; this is due to global character of the perturbation in (4).
In the same spirit, the propagation speed properties of classic KPP-Fisher equation, i.e., the equation (1) with h = 0, turn out to be rather different to h > 0. For instance, the seminal result of convergence in form given by Bramson [2] is not true, which is a topical issue [6,9,16]. More precisely, Proposition 1.2 Let h ≥ 1.87 and c ≥ 2. If u 0 is an initial datum as Proposition 1.1 then there is no a translations family β such that The proof is simply to replace in (6) the quantity −ct by β(t) and the right side by a suitable o(t) function, then the Proposition 1.2 is a consequence of the existence of proper semi-wavefronts (semi-wavefronts are unique up to translation, by [20]) when h ≥ 1.87 [7] and Proposition 1.1. So that, despite the stability results given in [1] and [21], the functional KPP-Fisher equations manifest some instabilities in a basic level.
Accordingly, the purpose of this work is to extend the stability results given in [1] to (1) and to give the first result to the stability of semi-wavefronts to the Neutral KPP equation (2). To the best of our knowledge, there is no published works about the stability of semi-wavefronts to equation (2). In a pioneer work [12] Liu and Weng proved the existence of a minimal speed to the existence of monotone wavefronts for neutral partial differential equations satisfying a standard monotony condition which (2) does not hold. Subsequently, the existence of monotone wavefronts to equation (2) was established by Hernández and Trofimchuk in [10] and [11]. Liu [13] proved that two non critical monotone wavefronts to (2) are the same up to a translation, then in [11] it was proved that if φ c is a monotone wavefront to (2) then another semi-wavefront with speed c is equal to φ c up to a translation.
On the other hand, Hasik et al [8] showed that equation (1) has semiwavefronts with speed c if and only if c ≥ 2. To the associated wave equation to (1), i.e., they demonstrated that a semi-wavefront with speed c is actually a wavefront when c > 2 ∞ 0 s Q(s)ds, which is a generalization of the result established in [7] for t > 0, x ∈ R; where µ 1 and µ 2 are Borel measures. As in [12], under the change of variable v(t, x) = u(t, x) − bu(t − h, x) equation (2) can be transformed to equation (10). More precisely, in the equation So that, our approach is based in an abstract result to partial functional differential equations with infinite delay, in a similar way of [18]. Also, this result includes the so called Mackey-Glass equations [14,15,21,17].
The paper is organized in the following way. In section 2 we present our main result which will prove in the section 3. In the section 4 we apply our abstract result to functional KPP-Fisher equations.

Main Result
In this section we consider a linear evolution system is jointly continuous in t and σ, S(σ, σ) = I and Henceforth, for u : R → R we denote u t as the element in C λ , define pointwise by u t (s) := u(t + s). The following result is obtained from [4] Proposition 2.1 (Existence of solutions) Suppose that S(·, ·) holds (S 1 ) and (S 2 ). Also, suppose F : [0, +∞) × X → X continuous satisfying, for all t ≥ 0, for all t ≥ 0 and z, z ′ ∈ X, then for each ρ ∈ C 0 the integral equation, for t ≥ σ, has a unique solution r : R → X which satisfies r t ∈ C 0 for all t ≥ σ.
Theorem 2.2 Let F and G i holding the conditions of Proposition 2.1 and Γ(s)e −αs dµ(s) < ∞ and define λ 0 as the only real solution of the following characteristic equation: Suppose that r,r ∈ C(R, X) satisfy the equation and In particular, if r σ ∈ C λ 0 in Proposition 2.1 then r t ∈ C λ 0 for all t ∈ [0, T ).

Proof of Theorem 2.2
In this section, for h > 0 we consider the equation (16) for t ∈ [σ, T ), and λ * = λ(h) ∈ R defined by the characteristic equation We define the operator for some b 0 ∈ R. If x(s) ≤ b 0 e σ(α−λ) e λ * s , for all s ≤ σ and λ * defined by (17), then The proof of Proposition 3.1 is a consequence of the following two lemma.
Therefore r n (t) converges to r * (t), uniformly on t ≤ T. Finally, the conditions on {S(·, ·)} on F and G i imply that r * is the solution of (12) with r * (s) = r 0 (s), s ≤ σ. So, the lemma has been proved. Otherwise, (13) is followed by the Lemma 3.6 and the Theorem 3.4 with h = h n , and (14) is a consequence of Lemma 3.5.

Stability of Traveling Waves
Now, we begin establishing an estimation of the linear part of a functional partial differential equation. For β ∈ R and x 0 ∈ R we define ω β (x) = min{e βx , 1} and the following Banach space We consider the evolution system S(t, τ ), t ≥ τ ≥ 0, associated to the following Cauchy problem with ρ ∈ X β . Henceforth, we assume that the initial data u 0 is Hölder in the x-variable in order to obtain the parameter variation formula associated to S(t, τ ) from the partial functional differential equation, cf. [5].
We denote In this section we will need the following result Proof: Denote the differential operator Lδ = Dδ xx − cδ x + mδ − δ t and, for is a continuos function on R which satisfies we can proceed as in [19, Proof of Lemma 1] in order to conclude that |ρ| X β < ∞ implies for all t ≥ τ and x ∈ R.

Hutchinson with infinite delay
We consider the KPP-Fisher with infinite delay where Q(s) ≥ 0 for all s ≥ 0 and +∞ 0 Q(s)ds = 1. By making the change of variable x → x + ct we have the equation (31) become Now, we fix a solution v(t, x) of (32) such that v 0 (s, x) ≥ 0 for all (s, x) ∈ (−∞, 0]× R, a simple application of the Maximum Principle yields to v(t, x) ≥ 0 for all Accordingly, µ is the Lebesgue-Borel measure, F (t, ξ, ζ)(x) = −v(t, x)ξ(x) and G 1 (s, z)(x) = Q(s)T cs z(x). Now, we take x 0 = +∞, namely, the weight e βx , therefore and Note that m 0 = 1 and Now, as in [21] we asume a bound to v(t, x), so there is M 0 > 0 such that |v(t, x)| ≤ M 0 for all (t, x) ∈ R ≥0 × R. Thus, we can apply the Theorem 2.2 and the characteristic equation is So that, the condition Γ(0) < −α is equivalent to Thus, we have the following result which includes all monotone wavefronts with speed c ≥ 2 √ 2.

Neutral Partial Differential Equations
We consider a general neutral partial differential equation where K > 0 and we assume a general hypothesis on f Under the monostability condition ∇f (0, 0, 0) > (0, 0, 0) and a monotony condition on f it was proved in [12] the existence of a minimal speed c * to the existence of monotone wavefronts.

Acknowledgments
It is a pleasure to thank Sergei Trofimchuk for many valuable discussions. This research was supported by Fondecyt (Chile) #11190350.

Declarations
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