Slowly oscillating wavefronts of the KPP-Fisher delayed equation

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x \nu +ct)>0,$ $ |\nu|=1, $ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in \R^m. \eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $\phi(+\infty)=1$) if $c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of some auxiliary one-dimensional map with the negative Schwarzian. The analysis of the fronts' asymptotic expansions at infinity is another key ingredient of our approach. It allows to indicate the maximal domain ${\mathcal D}_n$ of $(\tau,c)$ where the existence of non-monotone wavefronts can be expected. Here we show that the problem of wavefront's existence is closely related to the Wright's global stability conjecture.


Introduction and main results
The delayed KPP-Fisher equation or the diffusive Hutchinson's equation can be considered as one of the most important examples of delayed reactiondiffusion equations. In particular, during the past decade, this model has been studied by many authors, see [2,5,7,9,8,10,14,24] and the references therein. A significant part of the research dealt with the existence of traveling fronts connecting the trivial and positive steady states in (1) and in its non-local variant [3,6,11,23] u t (t, x) = ∆u(t, x) + u(t, x) We recall that the classical solution u(x, t) = φ(ν · x + ct), |ν| = 1, is a wavefront (or a traveling front) for (1) or (2) propagating at the velocity c ≥ 0, if the profile φ is non-negative and satisfies φ(−∞) = 0 and φ(+∞) = 1. By replacing condition φ(+∞) = 1 with less restrictive 0 < lim inf s→+∞ φ(s) ≤ lim sup s→+∞ φ(s) < ∞, we get the definition of a semi-wavefront. The nonnegativity requirement φ ≥ 0 is due to the biological interpretation of u as of the concentration of a dominant gene that is reminiscent of the seminal works by Kolmogorov, Petrovskii, Piskunov and Fisher.
Recently, the wavefront existence problem for (1), (2) was considered by using quite different approaches. The first method was proposed by Wu and Zou in [24]. It uses the positivity and monotonicity properties of the integral operator where (Hφ)(s) = φ(s)(b+1−φ(s−h)), h := cτ, is taken with some appropriate b > 1, and z 1 < 0 < z 2 satisfy z 2 − cz − b = 0. A direct verification shows that the profiles φ ∈ C(R, R + ) of semi-wavefronts can be also identified as positive bounded solutions of the integral equation Aφ = φ satisfying the above mentioned boundary conditions at ±∞. Unfortunately, the presence of positive delay in (3) strongly affects the monotonicity of A. In order to overcome this difficulty, two different orderings, the usual one and a nonstandard Smith and Thieme ordering of C(R, R + ), were combined in [24]. Even so the operator A was monotone with respect to each of these two orderings only for sufficiently small h and monotone φ.
The operator A is well defined when b > 0. Taking formally b = −1 in (3) and interpreting correctly the obtained expression for c > 2, instead of A we obtain (Bϕ)(t) = 1 µ − λ +∞ t (e λ(t−s) − e µ(t−s) )ϕ(s)ϕ(s − h)ds, where 0 < λ < µ are the roots of z 2 − cz + 1 = 0. Remarkably, all monotone wavefronts to equation (1) can be found via a monotone iterative algorithm which uses B (or its limit version B 2 if c = 2) and converges uniformly on R, see [10]. Similar ideas were also successfully applied in [5,6,14]. However, our attempts to use the monotone operator B in the case of non-monotone waves were not fruitful.
Aiming to get rid of monotonicity requirements, Shiwang Ma achieved an important progress in [16,17]. He showed that operators similar to A, B have good compactness properties in suitable Banach spaces. Therefore, in certain situations, the Schauder fixed point theorem could be used instead of the iterative monotone scheme from [24]. Ma's idea was successfully applied to various reaction-diffusion models with bounded nonlinearities. Nevertheless, equation Aφ = φ with A defined by (3) has never been considered within the Ma's approach: this is mainly because of the considerable difficulties related to the construction of a nontrivial A-invariant set suitable for the application of the Schauder fixed point theorem.
It is therefore tempting, in order to avoid the construction of a non-trivial bounded A-invariant convex closed set Ω, to apply the Leray-Shauder continuation principle to equation Aφ = φ. The main obstacle for the realization of such an idea is the apparent impossibility to have at the same time complete continuity of A and the non-empty interior of Ω. This problem was avoided in a nice way by Berestycki et al. in [3]. Working with equation (2), for a fixed δ > 0, Berestycki et al. considered a family of associated boundary value problems, with the boundary conditions φ n (−n) = 0, φ n (n) = 1, φ n (0) = δ. Fortunately, the above mentioned contradiction between the compactness of operator and the openness of its domain does not occur on finite intervals [−n, n]. Hence, the Leray-Shauder continuation principle (with corresponding calculation of a priori estimates, degrees etc) can be applied for each n ∈ N. Finally, the wave profile φ was obtained in [3] as the limit of φ n . The proof of the existence in [3] is rather technical and non-trivial. Regrettably, the conditions of C 1 -smoothness of kernel K and especially the positivity of K(0) > 0 do not allow use the existence theorem from [3] to deduce a similar result for equation (1). Indeed, if we take some δ−like sequence of kernels then the corresponding sequence of traveling waves φ (j) (s) could be eventually unbounded in view of a priori estimates obtained in [3].
Our short description of analytical tools used to prove the wave existence in (1), (2) would be incomplete without mentioning the Lin-Hale approach to heteroclinic solutions developed in [7,?]. This method allowed to obtain almost optimal existence results (i.e. τ ≤ 3/2 and c ≥ c ′ , for some indefinite and large c ′ : see also Fig. 1 and Conjecture 1 below) for rapidly traveling fronts. Nevertheless, the most interesting in applications critical waves were excluded in [7,?]. Surprisingly, as the recent work [9] shows, the Lin-Hale method still can be extended to give a complete solution to the problem of existence of monotone fronts in several models (including (1)). However, the monotonicity of waves is one of crucial assumptions in [9] and, at this moment, it is not clear whether it can be dropped.
After analyzing the above approaches to the existence problem and motivated by [3,16,24], we decided to work with the equation Aφ = φ. As a result, we elaborated a framework suitable for the application of the Schauder fixed point theorem for an appropriately modified version of the operator A. Before stating the corresponding existence theorem, let us define several subsets of parameters (τ, c) ∈ R 2 + (see also The proof of Theorem 1 requires a detailed study of oscillation/monotonicity properties of semi-wavefront profiles. Here we were inspired by geometrical descriptions from [22] of semi-wavefront profiles to the Mackey-Glass type delayed reaction-diffusion equation It is known that in the ordinary case (i.e. when u = u(t)) models (1), (5) can be considered within the same family of differential equations governed by linear friction (possibly, degenerate) and negative delayed feedback. Inclusion of the diffusive terms, however, makes the similarity between (1) and (5) much less direct. Nevertheless, it is still possible to prove that the semi-wavefront profiles to (1) share all geometric properties established in the case of equation (5). Amazingly, the statements of corresponding assertions become even sharper while their proof simplifies: cf. Theorem 2, 4 below with Theorems 1,3 in [22].
As in [22], we follow closely the definition of slow oscillations from [19,20]: we define the number of sign changes by We will say that ϕ(t) is sine-like slowly oscillating if graph of ϕ oscillates around 1 and has exactly one critical point between each two consecutive intersections with the level 1, and, in addition, for each t ≥ T 0 (T 0 was defined in Theorem 2), it holds that either sc(φ t ) = 1 or sc(φ t ) = 2.
Our next result is similar to [22,Theorem 3]. In fact, it is even stronger, since it excludes non-monotone but eventually monotone wavefronts to equation (1). As the numerical simulations of [3, Figure 1] show, this irregular behavior can occur in simple non-local KPP-Fisher equations. We also believe that such kind of irregular non-monotone wavefronts can be found in equation (5).
Remark 5 By Theorem 10 below, each bounded profile φ has to develop nondecaying slow oscillations around 1 for each c > c ⋆ (τ ) and then, due to [20], these oscillations should be asymptotically sine-like periodic.
The final part of this section concerns the determination of domain D n ⊂ R 2 + . We recall that D s was already found in Theorem 1 while the complete description of D m was given in [10]: where the convergence is monotone and uniform on R. Finally, for each fixed c = c * (τ ), φ(t) is the only possible monotone profile (modulo translation).
As it was recently demonstrated by Fang and Wu in [5, Theorem 6.2], condition c = c * (τ ) of Proposition 6 can be dropped. In any case, the uniqueness in [5,10] was established only within the class of monotone fronts (see also [6] for a similar assertion concerning (2)). Here, by combining the Berestycki-Nirenberg sliding argument [4] with the approach of [10], we obtain the following Theorem 7 Suppose that (τ, c) ∈ D m and u = φ 1 , φ 2 are wavefronts to (1).
The sliding solutions method does not work when (c, τ ) ∈ D m . However, as the recent works [1,8] have showed, the uniqueness (up to translation) of the semi-waveronts to (1) is very likely to be true for large speeds. We believe that for each fixed pair (τ, c) the semi-wavefront solution to equation (1) is unique (up to a translation) whenever it exists.
Theorem 7 is instrumental in proving Theorem 4 and, whence, in establishing our last two results: Moreover if (τ, c) ∈ D then necessarily φ(+∞) = 1. Hence, for each τ ≤ 1 equation (1) has at least one semi-wavefront which necessarily is a wavefront.
Theorem 10 (Admissible wavefront speeds and non-existence of fronts) Eq. (1) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c > c ⋆ (τ ) or c < 2.
It can bee seen from Proposition 6 and Theorem 10 that  Figure 1. In this way, considerations of the present work suggest the following natural criterion for the existence of non-monotone wavefronts in (1): It can be regarded as an extension of the famous Wright's global stability conjecture [13,15]. Therefore, in our opinion, it would be very interesting (and, perhaps, very difficult) to prove it. In particular, in the limit case c = +∞, Conjecture 1 is true if the Wright's conjecture is true. An important partial result in proving Conjecture 1 would be the following analog of the Wright's 3/2-global stability theorem: The structure of the remainder of this paper is as follows. Section 2 contains the proof of Theorem 7. In the third section, we describe the geometrical form of semi-wavefronts. Theorems 8, 1 and 10 are proved in Sections 4, 5, 6 respectively. In Appendix, the characteristic function of the variational equation at the positive steady state is analyzed.

Absolute uniqueness of monotone wavefronts
Take some (τ, c) ∈ D m . Then by Proposition 6 and [5, Theorem 6.2] there exists a unique monotone wavefront u = ψ 2 (ν · x + ct). Suppose that u = ψ 1 (ν · x + ct) is a different (and therefore non-monotone) wavefront. Clearly, each profile ψ i (t) satisfies and therefore it is strongly positive due to PROOF. Suppose that, for some s, we have φ(s) = 0. Since φ(t) ≥ 0, t ∈ R, this yields φ ′ (s) = 0. Therefore y = φ(t) satisfies the following initial value problem for a linear second order ordinary differential equation But then y(t) ≡ 0 due to the uniqueness theorem.

Lemma 13
Suppose that (τ, c) ∈ D m and let λ 0 < 0 be as in Lemma 30. Let c ∈ [2, c * (τ )), q ∈ R, and ǫ > 0 be sufficiently small. Then for some K 2 > 0 and K 1 ∈ R independent on q. Similarly, if c = c * (τ ) then PROOF. In the monotone case (i.e. i = 2), this statement follows from [10, Lemma 28] and Lemma 30 (see also [10,Theorem 6] for more details). Next, due to [10, Lemma 10], the condition (τ, c) ∈ D m implies the hyperbolicity of the positive equilibrium of (7) and therefore |ψ 1 (t)−1| converges exponentially to 0 at +∞. With this observation, the analysis of the non-monotone wavefront is completely analogous to the monotone case considered in [10,Section 7]. The unique exception is the sign of K 1 . Indeed, in virtue of non-monotonicity of the wavefront ψ 1 , K 1 could take any real value including 0.
By applying a sliding argument, we are ready now to prove Theorem 7. Set It follows from Lemmas 12, 13 that Q = ∅. On the other hand, it is obvious that the set Q is closed, below bounded and connected (the latter is due to the monotonicity of ψ 2 ). Let q * = inf Q, we claim that, for some finite t * , Indeed, otherwise and therefore Lemma 12 (taken with q = 0 and applied to ψ 1 and ψ 3 ) implies that there are S 0 and δ 0 > 0 such that . Now, applying Lemma 13 (with q = 0) to the profiles ψ 3 and ψ 1 we obtain that necessarily K 2 ≥ K 1 . We claim that K 2 > K 1 . Indeed, otherwise K 2 = K 1 > 0 and therefore the uniqueness proof of [10, Section 6.3] can be repeated for c ≤ c * (τ ), see also [5,Theorem 6.2]. Hence, K 2 > K 1 and therefore there exist Finally, considering inequality (9) on a fixed interval [S 0 , S 1 ], we find that, for some Therefore q * − δ * ∈ Q, a contradiction.
Hence, (8) holds and therefore non-negative function θ(t) = ψ 1 (t) − ψ 3 (t) attains its zero minimum at t * . Moreover, as θ(t) > 0 for t ≤ S 0 , we may assume that t * is the leftmost zero minimum of θ. Now, it is easy to see that bounded θ also satisfies the differential equation so that either Considering the above relations with t = t * , we deduce immediately that Θ(s) ≡ 0 on [t * , +∞). However, this can not happen because of the inequality The obtained contradiction ends the proof of Theorem 7. By Lemma 11, similarly to the case of the Hutchinson's equation, the change of variables φ(t) = e −x(t) can be applied to (7). The obtained equation (see equation (10) below) is a unidirectional monotone cyclic feedback system with delay [20]. Therefore, analogously as it was done in [22], fundamental results from [19,20] can be used to demonstrate slowly oscillating character of the non-monotone semi-wavefronts. Nevertheless, here we have preferred to give short and self-contained direct proofs of this fact, additionally establishing sinusoidal shape of all (and not only periodic as in [20]) oscillating solutions. See also Remark 5 in the introduction.
< 0 for all t > s, and therefore φ(t) can not be positive for large t, a contradiction.

Lemma 15
Let Q 0 be as in Lemma 14 and Q 1 be such that φ(s) > 1 for all s from some maximal open interval (Q 0 , Q 1 ). Then the only options for the geometrical shape of φ on (Q 0 , Q 1 ) are: (II) φ strongly increases on (Q 0 , +∞), with at most one critical point Q 0 + h. (III) φ has exactly two critical points: strong local maximum at T 0 ∈ (Q 0 , Q 0 + h) and a strong local minimum at PROOF. Obviously, we get the second option if φ ′ (t) > 0 for all t ∈ R. Thus we may suppose that there exists some leftmost point T 0 > Q 0 where φ ′ (T 0 ) = 0. This implies immediately that φ(T 0 ) > 1, φ ′′ (T 0 ) ≤ 0, and, consequently, φ(T 0 − h) ≤ 1.
First, assume that φ ′ (t) > 0 for t > T 1 . Then φ(t) is unbounded since otherwise φ(t) converges monotonically to 1 that is possible only when (τ, c) ∈ D m and therefore this contradicts to Theorem 7. As a consequence, there exists a finite Q 2 with the mentioned properties.
Suppose now that there exists some leftmost point ). This means that φ ′′ (T 2 ) < 0 and φ(T 2 − h) < 1. But then φ can not have any critical point b > T 2 , φ(b) < 1, since otherwise we get a contradiction: can not be positive for large positive t. The latter contradiction shows that actually T 2 > Q 2 and thus Q 2 is finite and φ ′ (Q 2 ) > 0. Finally, Q 2 − Q 0 > T 1 − Q 0 > h while the inequality T 2 − h < Q 2 can be proved in the same way as the inequality T 0 − h < Q 0 in Lemma 15(I).
Corollary 18 Graph of each oscillating solution consists from the arcs similar to described in Lemmas 15(I),17 and therefore it is sine-like slowly oscillating.
Our a priori estimates are based on the following key assertion: Similarly, g(s).
Next, it is clear that γ ≥ 0. If γ = 0 then B ≥ g(a) = −y ′ (a) ≥ 0 and the claimed inequality is immediate. If γ > 0 then γ ∈ {λ 1 (s ′′ ), λ 2 (s ′′ )} for some s ′′ ∈ (a, b). As a consequence, we obtain the second estimation of the lemma: Recall that the Schwarz derivative Sp of C 3 -smooth function p is defined as x ∈ R, (f • g)(0) = 0, is well defined, strictly decreasing and has the negative Schwarz derivative on R.

Lemma 22
Let c ≥ 2 and φ(t), φ(−∞) = 0, be a slowly oscillating on [Q 0 , +∞) positive solution of equation (7). Then φ is bounded and where PROOF. Without the loss of generality, we can set Q 0 = 0. Then it suffices to prove the boundedness of x(t) = − ln φ(t) on [0, +∞). Since φ(t) is slowly oscillating about 1, the transformed solution x(t) oscillates slowly around the zero equilibrium of (10). This implies that there exists an increasing sequence where w(x) := e −x − 1. Next, consider V 2 = x(T 2 ) < 0, we have x ′ (T 2 ) = 0, x(Q 2 ) = 0 and T 2 − Q 2 < h. Recalling that φ(t) (and, consequently, x(t)) is sine-like slowly oscillating (so that x ′ (t) < 0 on (T 1 , T 2 )) and applying Lemma 20, we obtain Since Q j+2 − Q j > h for each j, we can repeat the above two steps to conclude that As a consequence, and therefore, after setting L(c, h) = min {−ch, B * (c, h)}, we obtain that This ends the proof of Lemma 22.
whose coefficient a(t) = 1 − φ(t − h) is uniformly bounded on I by a constant depending only on c, h. We consider separately the cases m = 0 and m > 0.
Now we can assume that m > 0 and φ(t) < 1 on some maximal interval (Q m−1 , Q m ). We also know that φ ′ (t) > 0 on some maximal open subinterval (T m−1 , Q m ) of (Q m−1 , Q m ). Since φ ′ (T m−1 ) = 0, φ(T m−1 ) < 1, we can integrate equation (12) repeatedly (as it has been done in the case m = 0) to prove the existence of Step II. Suppose now that S m = T m ≥ Q m . This situation corresponds to the cases (II) and (III) of Lemma 15. Since φ ′ (S m ) = 0 and φ(t) ≤ U e (c, h), t ≤ S m , we can again integrate equation (12) repeatedly to prove the existence of For fixed c ≥ 2, h > 0, we will consider also the following modified equation with β(c, h) defined in Corollary 23 and with continuous piece-wise linear max{0, 2β(c, h) − u}, u > β(c, h). (13) and (7) share the same set of semi-wavefronts.
PROOF. Due to Lemma 26, it suffices to prove the first inequality in (19) for ). In order to evaluate Q(t), we consider the following chain of inequalities (for t ≤ T c ) But then, rewriting the latter differential inequality in the equivalent integral form (e.g. see [16]) and using the fact that and Lemma 27 is proved.
Finally, it is clear that, in order to establish the existence of semi-wavefronts to equation (14), it suffices to prove that the equation A m φ = φ has at least one solution from the subset is defined with some fixed ρ > 0. Observe that the convergence x n → x on K is equivalent to the uniform convergence x n ⇒ x on compact subsets of R.
Lemma 28 Take c > 2. Then K is a closed, bounded, convex subset of X and A m : K → K is completely continuous.
PROOF. By the previous lemma, A m (K) ⊂ K. It is also obvious that K is a closed, bounded, convex subset of X. Since due to the Ascoli-Arzelà theorem A m (K) is precompact in K . Next, by the Lebesgue's dominated convergence theorem, if Hence, the map A m : K → K is completely continuous.
Theorem 29 Assume that c ≥ 2. Then the integral equation A m φ = φ has at least one positive bounded solution in K.
PROOF. If c > 2 then, due to the previous lemma, we can apply the Schauder's fixed point theorem to A m : K → K. Let now c = 2 and consider c j := 2 + 1/j with h 0 := 2τ, h j := c j τ . By the first part of the theorem, we know that for each c j there exists a semi-wavefront φ j : we can normalize it by the condition φ j (0) = 1/2, φ ′ j (s) > 0, s ≤ 0. It is clear from (20) that the set {φ j , j ≥ 0} is precompact in K and therefore we can also assume that φ j → φ 0 in K, where φ 0 (0) = 1/2 and φ 0 is monotone increasing on (−∞, 0]. In addition, R j (s) := r(φ j (s), φ j (s − h j )) → R 0 (s) := r(φ 0 (s), φ 0 (s − h 0 )) for each fixed s ∈ R. The sequence {R j (t)} is also uniformly bounded on R. All this allows us to apply the Lebesgue's dominated convergence theorem in where z 1,j < 0 < z 2,j satisfy z 2 − c j z − b = 0. In this way we obtain that A m φ 0 = φ 0 with c = 2 and therefore φ 0 is a non-negative solution of equation (7) satisfying condition φ 0 (0) = 1/2 and monotone increasing on (−∞, 0]. It is immediate to see that φ 0 (−∞) = 0 and therefore φ 0 is a semi-wavefront.

Admissible wavefront speeds
First, we observe that the necessity of the condition c ≥ 2 for the existence of monotone wavefronts was already established in [10, Lemma 19]. Since the leading edge of each semi-wavefront is monotone, the proof of the mentioned lemma is also valid for the broader family of semi-wavefronts.
Consider now some semi-wavefront φ propagating at the velocity c > c ⋆ . We know that φ is slowly oscillating around the positive steady state. In this section, we show that these oscillations are non-decaying.
Thus y n (s − h) > 0 that yields s − h > −r n . Consequently,s − h > −r n for each other critical points of y ′ n (t). All this implies that |y n (s − h)| ∈ [0, 1]. Therefore |y ′ n (t)| ≤ 1.1/c for t ≥ 0, and, in particular, y n (t) ≥ 0.45 on [0, c/2]. Next, due to the Ascoli-Arzelà theorem, the sequence y n (t) has a subsequence which converges on [0, +∞), in the compact-open topology, to some continuous function y * (t). Evidently, max{|y * (s)|, s ≥ 0} = y * (0) = 1 and y * (t) ≥ 0.45 on [0, c/2]. Next, for some fixed positive b and all t ∈ [h, +∞), it holds that In order to establish some further properties of y * (t), let us present the family of all solutions to (21) which are bounded at +∞: Here ǫ ′ = z 2 − z 1 is defined in the same way as in Lemma 26. Replacing y(t) with y n (t) in (22), we obtain that, for some A n , The latter inequality implies that A n , n ∈ N, are uniformly bounded: Hence, taking limit as n → +∞ (through passing to a subsequence if necessary) we find that y * (t) satisfies with some finite A. Now, (23) implies that y * (t) is a solution of the equation We claim that y * (t) is not a small solution. Indeed, on the contrary, let us suppose that y * (t) has superexponential decay. Since the characteristic function z 2 − cz − e −zh to (24) has the exponential type h, an application of [12, Theorem 3.1] assures that y * (t) = 0 for all t ≥ 2h. But then equation (24) implies that y * (t) = 0 for all t ≥ h and, in consequence, y * (t) = 0, for all t ≥ 0. This contradicts the inequality y * (t) ≥ 0.45 on [0, c/2] and therefore y * (t) is not a small solution.
This implies the existence of an interval (a, a + h), a > 3h, such that y * (t) changes its sign on (a, a+h) exactly three times. Since y n j (t) → y * (t) uniformly on [a, a + h], we can conclude that sc(ȳ n j ,a+h ) ≥ 3 for all large j. However, this contradicts to the slowly oscillating behavior of y n j (t). In consequence, the equality φ(+∞) = 1 can not hold for c > c ⋆ .
In this section, we study the zeros of ψ(z, c) := z 2 − cz − e −zcτ , c ≥ 2, τ > 0. It is straightforward to see that ψ always has a unique positive simple zero λ −1 . Since ψ ′′′ (z, c) is positive, ψ can have at most three (counting multiplicities) real zeros, one of them positive and the other two (when they exist) negative.