Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h>0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.


1.
Introduction. In this work, we focus our efforts on the study of the existence, uniqueness and asymptotics of positive monotone bounded traveling wave solutions u(t, x) = φ(ν · x + ct), φ(−∞) = 0, to the scalar reaction-diffusion equation u t (t, x) = ∆u(t, x) − u(t, x) + g(u(t − h, x)), x ∈ R m . (1) It is assumed that ν ∈ R m , |ν| = 1, that the wave velocity c is positive and the continuous monotone nonlinearity g : R + → R + satisfies the following assumption (H) g is strictly increasing and the equation g(x) = x has exactly two nonnegative solutions: 0 and κ > 0. Moreover, g is differentiable at the equilibria with g ′ (0) > 1, g ′ (κ) < 1, and g is C 1 -smooth in some neighborhood of κ. In addition, there exist C > 0, θ ∈ (0, 1], δ > 0 such that Perhaps, model (1) is one of the simplest and most studied monostable delayed reaction-diffusion equations. See [1,5,15,16,21,22,25,32,34,33,35,37] and references therein for more detail regarding (1) and its non-local versions. In fact, the last decade of studies has lead to almost complete description of the existence, uniqueness and stability properties of wavefronts to (1) whenever g satisfies (H) and the following quite important sub-tangency condition The latter inequality was already used in the celebrated work [20] by A. Kolmogorov, I. Petrovskii and N. Piskunov, where it was assumed that g ′ (x) < g ′ (0) for all x ∈ (0, κ]. Roughly speaking, inequality (3) amounts to the dominance of the 'linear component' within essentially non-linear model (1). It is needless to say that, from the technical point of view, (3) allows to simplify enormously the analysis of traveling waves. In Subsections 1.1-1.3 below we will illustrate this point in greater detail by discussing such key issues as the minimal (critical) speed of propagation, the stability, existence and uniqueness of waves, the asymptotic properties of wave profiles. Therefore it is not a big surprise that none of these issues has been adequately addressed in that strongly nonlinear case when (3) does not hold and h > 0 1 . So our main objective in this paper is to complete the study of the existence, uniqueness and asymptotics of wavefronts to delayed reaction-diffusion equation (1) considered under hypothesis (H) and without condition (3). At this stage of discussion, it is instructive to raise the same questions but for a different family of delayed evolution equations It is obtained from (1), m = 1, by a formal discretization of the Laplace operator. Equivalently, we can consider the lattice differential equations u ′ n (t) = [u n+1 (t) + u n−1 (t) − 2u n (t)] − u n (t) + g(u n (t − h)), n ∈ Z.
Equations (4), (5) are special cases of more general nonlocal lattice population model proposed in [36]. These equations were analyzed by Ma and Zou in [23]. Once again, in order to prove the existence, uniqueness, monotonicity and stability of wavefronts, (3) together with (H) were assumed in the cited work. One of the notorious features of [23] consists in its novel (and non-trivial) proof of the wave uniqueness. This proof does not impose any restriction on sup{g ′ (x), x ∈ [0, κ]} what is remarkable in the case of delayed equations, cf. Subsection 1.2.
On the other hand, starting from the pioneering work of Zinner, Harris and Hudson [39], significant progress has been achieved in the understanding of waves solutions in non-delayed versions of (4), (5). See [8,9,23,39] for more information and further references. Non-delayed equation (4) can be also viewed as a particular case of the following differential equation with convolution which was firstly introduced by Kolmogorov et al in [20]. The latter equation was thoroughly investigated during the past three decades using various techniques, see [1,7,10,11,30] and references therein. Remarkably, sub-tangency condition (3) was avoided in the recent important contributions [8,9] by Chen et al. and [10,11] [31]. As in [8], we construct a new formal upper solution (for some velocity c ′ close to a given velocity c) from a given wavefront φ(t, c). However, in difference with [8], our upper solution is not only formal but also true upper solution appearing in pair with an appropriate lower solution. We neither apply the truncation procedure as in [8,11,39] nor we use our upper solution as a bound obligating solutions of associated truncated problems to converge to a true wave solution (this nice idea was proposed in [8]). We consider φ(t, c) only as a skeleton (we call it 'a base function') for creating a true upper solution by its suitable modification. Recently, the method of base functions was successfully applied in our previous work [31] to a model of the Belousov-Zhabotinskii reaction. Now, two noteworthy differences appear while comparing (6) and (1). First of them is technical: the presence of the second derivatives in (1) complicates the construction of the lower and upper solutions for (1) (these solutions must be C 1smooth or satisfy additional conjugacy relations at the discontinuity points of the derivative , cf. [3,6,31,37]). The other difficulty is more essential: the presence of positive delay h can lead to the non-monotonicity of traveling fronts [4,16,26,32,34] while such monotonicity seems to be crucial for the applicability of various approaches, e.g. of the sliding solution method [3,9,10,11]. Precisely in order to avoid front oscillations around κ, we will consider strictly increasing g in (H). It should be mentioned that monotonicity of g is not obligatory when h = 0: this is because function g(u(t − h)) + ku(t) is monotone in u(t) for k ≫ 1, h = 0, cf. [1].
Before going back to more detailed analysis of the main problems addressed in this paper, we would like to state some useful results concerning the wavefronts to equation (1) considered under assumption (H). Set g ′ + := sup x≥0 g(x)/x ≥ g ′ (0) > 1 and define c # [respectively, c * ] as this unique positive number c for which the characteristic equation with p = g ′ (0) [respectively, with p = g ′ + ] has a double positive root. It is easy to see that c # ≤ c * . Note that c # = c * coincides with the minimal speed of propagation c * whenever (3) is satisfied. If c > c # then the characteristic equation (7) with p = g ′ (0) has exactly two real solutions 0 < λ 2 < λ 1 , λ j = λ j (c). Finally, if c = c # , then the following asymptotic representation is valid (for an appropriate s 0 , j ∈ {1, 2} and some σ > 0): If c = c # then besides (8) it may happen that Proof. The existence of fronts for c ≥ c * follows from [33,Theorem 4] while their non-existence for c < c # is a well known fact (e.g. see [33,Theorem 1]). Due to [32,Corollary 12], the wave profiles ψ are monotone, with ψ ′ (s) > 0, s ∈ R. The exponential convergence ψ(t) → 0, t → −∞, is a consequence of the Diekmann-Kaper theory, see [12] and [1, Lemma 3]. Therefore there is δ > 0 such that On the other hand, it is easy to see that the convergence ψ(t) → 0, t → −∞, is not super-exponential, cf. [ (3), the minimal speed of propagation c * can be computed from the characteristic equation (7) considered with p = g ′ (0). Without (3), the computation of c * represents a very difficult task even for non-delayed models [2,17,38]. In such a case, the value of c * depends not only on g ′ (0) but also on the whole nonlinearity g. Furthermore, if h > 0 and (3) does not hold, the situation becomes even more complicated: it is an open question whether there exists a positive c * splitting R + on subsets of admissible and non-admissible (semi-) wave speeds. In the present paper, we answer positively this question at least for g satisfying assumption (H) with (2) replaced with the slightly more restrictive inequality If g is not monotone, the existence of such c * remains an unsolved problem. In any case, for non-monotone g, it is necessary to introduce some adjustments to the definition of traveling front solution, replacing it with the concept of semi-wavefront solution, see [32,34].
1.2. Uniqueness of wavefronts. More subtle aspects of uniqueness and stability of wavefronts in (1) were studied so far under the geometric conditions even more restrictive than (3). For example, g ′′ (s) ≤ 0 was required in the main stability theorem of [25]. Similarly, uniqueness (up to a shift) of each non-critical (i.e. c = c * ) monotone traveling front of equation (1) can be deduced from [35,Corollary 4.9] whenever g meets the conditions: Let us show that (A3) is stronger than (3). Indeed, after dividing the latter inequality by θ and taking limit as θ → +0, we find that Therefore g ′ (v) < g(v)/v, v ∈ (0, κ], that, after an easy integration, yields It is clear that the above inequalities are stronger that the Lipshitz condition which in turn is more restrictive than (3). Inequality (11) is one of the basic conditions of the uniqueness theory developed by Diekmann and Kaper, cf. [12] and [1]. Suppose, for instance, that g ∈ C 1,q in some neighborhood of 0. Then (11) implies the uniqueness of all non-critical [12] as well as critical [1] wavefronts to (1). Additionally, [1] establishes the uniqueness of all fronts propagating at the velocity c > c u where c u can be computed (similarly to c * in the sub-tangential case) from the equation An alternative approach to the uniqueness problem is based on the sliding method developed by Berestycki and Nirenberg [3]. This technique was successfully applied in [8,10,11,23] to prove the uniqueness of monotone wavefronts without imposing any Lipshitz condition on g. 2 In the present paper, inspired by a recent Coville's work [10], we use the sliding method to prove the following assertion: We note that, when h > 0, we were not able to drop the condition of strict monotonicity on g imposed in Theorem 1.2 (even while considering only monotone wavefronts). If h = 0, the monotonicity of g is not obligatory.
Remark 2. In various aspects, Theorem 1.3 improves and generalizes on the nonlocal case the main uniqueness theorem from [23]. In difference with the mentioned result, we do not impose sub-tangency condition (3) and we admit critical (minimal) waves. Next, the uniqueness result of [23] is valid only for profiles having prescribed asymptotic behavior at −∞. Note also that our proof is rather short and does not use the monotonicity of profiles. Now, condition c = 0 seems to be essential: [11,Proposition 6.7] suggests the possibility of infinitely many wave solutions (perhaps, discontinuous) for c = 0. On the other hand, Theorem 1.3 complements the main result of [14] (which is valid only for non-critical waves), where (11) was assumed together with the symmetry β(k) = β(−k). Even though [14] (see also [1] for several improvements) allows to consider non-monotone nonlinearity g.

1.3.
Asymptotic formulas for the wave profiles. It is well known [13] that in non-delayed case each critical wavefront which propagates at the velocity c * > c # (i.e. so called pushed wavefront) has its profile converging to 0 more rapidly than the near (i.e. propagating with the speeds c ≈ c * ) non-critical wavefront profiles. This contrasts with the case c * = c # , when the profile of the critical front (so called pulled wavefront) converges to 0 approximately at the same rate as the profile of each near wavefront does. Similar asymptotics were also established for wavefront solutions of lattice equation (4) without delay, see [9,Theorem 3]. Our third main result shows that the pushed fronts to (1) obey the same principle: , is a traveling front to equation (1). Then the following asymptotic represantions are valid (for an appropriate s 0 and some σ > 0): The proof of the second formula is the most difficult part of this theorem. In order to establish that the pushed fronts to (1) satisfy 2), it suffices to show that each wavefront having asymptotic behavior as in 1) is 'robust' with respect to small perturbations of the velocity c. This would imply the existence of wavefronts propagating at the velocity c ′ < c * provided that the critical front behaves as in 1). The necessary perturbation result is demonstrated here with the use of upper-lower solutions method. Note that, due to the use of a discontinuous upper solution, application of this method in the paper is not at all standard.
Hence, we have proven that there exists T 2 such that, for all σ close to 1, < 0, we conclude that E(t, σ) < 0 for all t ∈ R and σ close to 1.
Step II (Construction of an upper solution).
For a : , for some positive ν, C j and negative T 4 (which does not depend on b). Since χ(c ′ , λ 2 ) < 0, we may choose T 4 is such a way that E + (t, b) < 0 for all t ≤ T 4 , b ∈ (0, 1]. On the other hand, we know that, uniformly on each compact interval, E + (t, b) → E(t, σ), b → 0+. Therefore E + (t, b) < 0 for all t ≤ T 3 (0+) + 1 for all sufficiently small b. Now, let us define an upper solution φ + by φ + (t) := min{κ, φ b (t)}. It is clear that φ + (t) is continuous and piece-wise C 1 on R, being t 0 := T 3 (b) the unique point of discontinuity of the derivative where ∆φ ′ Step III (Construction of a lower solution). Consider the following concave monotone linear rational function and set g − (x) := min{g(x), p(x)}. It is clear that g − is continuous and increasing and that Moreover, in some right neighborhood of 0, function g − (x) meets the smoothness condition of (H). This implies the existence of a monotone positive function Step IV (Iterations). Comparing asymptotic representations of monotone functions φ − (t) and φ + (t) at −∞, we find easily that for some appropriate s 1 . Simplifying, we will suppose that s 1 = 0. In the next stage of the proof, we need the following simple result: Lemma 2.1. Let ψ : R → R be a bounded classical solution of the second order impulsive equation where {t j } is a finite increasing sequence, f : R → R is bounded and continuous at every t = t j and the operator ∆ is defined by ∆w| tj := w(t j +) − w(t j −). Assume that equation z 2 − cz − 1 = 0 has two real roots ξ 1 < 0 < ξ 2 , ξ j = ξ j (c). Then Proof. See [31]. Alternatively, it can be checked by a straightforward substitution that ψ defined by (14) verifies equation (13).
Similarly to [37], we also consider the monotone integral operator where ξ ′ j := ξ j (c ′ ). Using properties of functions φ − (t) and φ + (t), we deduce from Lemma 2.1 that The latter implies (see [37] for more detail) the existence of a monotone solution φ(t) such that This amounts to the existence of a wavefront propagating at velocity c ′ . Moreover, the latter estimations shows that, for some s 0 and positive δ, Now, if a * = 0 then ψ(t) ≥ φ(t), t ∈ R. We claim that, in fact, ψ(t) > φ(t), t ∈ R. Indeed, otherwise we can suppose that T is such that φ(T ) = ψ(T ). In this way, the difference ψ(t) − φ(t) ≥ 0 reaches its minimal value 0 at T , while ψ(T − ch) > φ(T − ch). But then we get a contradiction: In this way, Lemma 3.1 is proved when a * = 0 and consequently we may assume that a * > 0. Let σ > 0 be small enough to satisfy max s∈[κ−σ,κ] g ′ (s) ≤ 1.
Case I. First, we assume that T is such that, additionaly In such a case non-negative function reaches its minimal value 0 at some leftmost point t m , where Since ψ(t m ) < φ(t m ), we have that t m > T , so that In consequence, for some θ ∈ (κ − σ, κ), a contradiction. Observe that the strict inequality in the last line can be explained in the following way. The sign "≥" can be replaced with "=" in if and only if g ′ (θ) = 1 and ψ(t m − ch) − φ(t m − ch) = −a * . This, however, is impossible due to the definition of t m as the leftmost point where w(t m ) = 0. Case II. If (18) does not hold, then, due to the convergence of profiles at +∞, we can find large τ > 0 and T 1 > T such that Therefore, in view of the result established in Case I, we obtain that Define now τ * by Since, in addition, we conclude that ψ(t + τ * ) > φ(t), t ∈ R, cf. (17). Now, if τ * = 0, then Lemma 3.1 is proved. Otherwise, τ * > 0 and for each ε ∈ (0, τ * ) there exists a unique T ε > T such that It is immediate to see that lim T ε = +∞ as ǫ → 0+. Indeed, if T εj → T ′ for some finite T ′ and ε j → 0+, then we get a contradiction: ψ(T ′ + τ * ) = φ(T ′ ). Therefore, if ε is small, then that is ψ(t + τ * − ε) and φ(t) satisfy condition (18) required in Case I. Thus we get ψ(t + τ * − ε) > φ(t) for all t ∈ R, a contradiction to the definition of h * . This means that τ * = 0 and the proof of Lemma 3.1 is completed.
For a fixed c ≥ c * , both φ and ψ have the same type of asymptotic behaviour at −∞ described in Proposition 1.
As we have mentioned,χ(z, c) is analytic in the domain Π α , while Φ(z) = D(z)/χ(z, c) is analytic in ℜz ∈ (0, λ) and meromorphic in Π α . In virtue of (22), we can suppose that Φ(z) has a unique singular point λ in Π α which is either simple or double pole. Now, for some x ′′ ∈ (0, λ), using the inversion theorem for the Laplace transform, we obtain that If x ∈ (λ, (1 + θ)λ) then Observe also that functionχ(z, c) does not have zero other than λ in a small strip centered at ℜz = λ. Therefore

Since
Res z=λ e zt D(z) Res z=λ e zt D(z) , if χ ′ (λ, c) = 0, we get the desired representation. It should be noted here thatχ ′′ (λ, c) < 0, that Indeed, if the latter residue were equal to 0, then Φ(z) would not have a pole at λ. Finally, it is easy to check that cϕ Next, we claim that the statement of Lemma 3.1 is also valid for solutions of (21). Regardless the fact that we do not know either wavefronts are monotone on whole real line or they are not, the proof of Case II can be repeated almost literally. The monotonicity of wavefronts on (−∞, ρ) will be sufficient for this purpose. For instance, let us prove the following Proof. Due to the monotonicity of φ and ψ at −∞, we find that for every τ ≥ 0 there exists T (τ ) such that Let us prove that T (τ ) is bounded from below on R + . Indeed, otherwise there exists a converging sequence τ j such that T (τ j ) → −∞. In turn, this forces T (τ j ) + τ j → −∞. But then we may use the monotonicity properties of φ, ψ in order to get a contradiction: φ(T (τ )) = ψ(T (τ ) + τ ) > ψ(T (τ )). Since φ(s) < κ, s ∈ R, we deduce in a similar way that the sequence {T (τ j )} can not have a finite limit as τ j → +∞. Thus T (τ ) → +∞ as τ → +∞. Since φ(+∞) = ψ(+∞) = κ, the remainder of the proof is straightforward. Now, the following main changes should be introduced in the proof of Lemma 3.1: 1. Set ∆(t) = ψ(t) − φ(t). Instead of (17), we then have that ∆(T ) = ∆ ′ (T ) = 0, Here (non-strict) monotonicity of g is sufficient because of 2. If a * > 0, we take small positive σ > 0 and integer N 1 > 0 such that g ′ (s)) and then we assume additionally that T is such that 3. Similarly, in (19), the expression g(ψ(t m −ch))−g(φ(t m −ch)) should be replaced with g ′ (s)).
As a result, we get again a contradiction: To finalize the proof of Theorem 1.3, it suffices to repeat the last two paragraphs of the third section.

5.
Proof of Theorem 1.4. In virtue of the front uniqueness, the first statement of Theorem 1.4 was already proved in the previous section (cf. (15)) so we have to consider the case c = c * only. Suppose, contrary to our claim, that (without restricting the generality, we can assume that s 0 = 0), take some c ′ < c * close to c * and consider the following piecewise continuous function For sufficiently large M and small ǫ > 0, the above definitions yield large negative T 1 = T 1 (M, a) and large positive T 2 (ǫ). Therefore, if M is sufficiently large and a, c * − c ′ > 0 are sufficiently small, then we can suppose that, for all t ≤ T 1 , 0.5M e ρt χ(ρ, c ′ ) + C 1 aM e λ ′ 2 (1+θ)t < M e ρt (0.5χ(ρ, c ′ ) + C 1 a) < 0. Moreover, since ρ > λ 2 , we also can choose T 1 ≫ a in such a way that . Indeed, we can first determine (large negative)T 1 as the leftmost root of equation φ(t) = M e ρt (with M large and positive). This corresponds to the limit case a = 0. The inequality φ ′ + (T 1 −) > φ ′ + (T 1 +) is obvious in such a case. To prove the second inequality, suppose that for a moment that, for some S ∈ [T 1 − h,T 1 ), M e ρS = φ(S), M e ρt < φ(t), t ∈ (S,T 1 ).
Then ρM e ρS ≤ φ ′ (S) so that (assuming that M is large) a contradiction. Since a ≪ 1 can considered as a small perturbation parameter, we deduce that the mentioned properties hold for all small a (where T 1 is close toT 1 ).
Next, for t ∈ [T 1 , T + 1 ] and small positive ǫ, c * − c ′ , we have that Therefore, for all t ∈ R and small c * − c ′ , ǫ > 0, To finalize the proof of Theorem 1.4, it suffices now to repeat Steps III and IV of Section 2. The construction of an lower solution is possible because of c * > c # : this inequality assures the existence of two positive real roots λ 2 (c ′ ) < λ 1 (c ′ ) for all c ′ close to c * .