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Quantifying the threshold phenomena for propagation in nonlocal diffusion equations

We are interested in the threshold phenomena for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such phenomena. The outcomes dramatically depend on the tails of the dispersal kernel and can take a large variety of different forms. The strategy is to combine sharp estimates of the tails of the sum of i.i.d. random variables (coming, in particular, from large deviation theory) and the construction of accurate sub-and super-solutions.


Introduction
In this work we consider the solution u = u(t, x) of the one-dimensional integro-differential equation ∂ t u(t, x) = R J(y) u(t, x − y) − u(t, x) dy + f u(t, x) , t > 0, x ∈ R, (1.1) supplemented with some compactly supported initial data u(0, x) = u 0 (x). The function J : R → R is a nonnegative dispersal kernel of total mass 1 allowing, in population dynamics models, to take into account long distance dispersal events. The nonlinearity f = f (u) encodes the demographic assumptions and, in this manuscript, we focus on the case where it is of the bistable or of the ignition type. Precise assumptions will be given later on. Our concern is to understand the threshold phenomenon for propagation. As far as we know, we provide the first quantitative estimates for such phenomenon in (1.1).
The solutions of (1.1) share some properties with the ones of the local diffusion equation In particular both problems exhibit a threshold phenomenon, meaning that "small" initial data lead to extinction, whereas "large" initial data lead to propagation. In the local diffusion case (1.2), we refer to [4] for such a property, while the sharp threshold phenomenon was more recently investigated [29], [13], [27], [23,24], through different technics. On the other hand, the nonlocal diffusion case (1.1) is more delicate, see subsection 3.4 below, but some progresses were recently achieved [1], [6], [20]. Very recently, the authors (including the first two authors of the present work) of [3] have provided a sharp quantitative estimate of the threshold phenomenon in (1.2). To be more precise, let f = f (u) be of the ignition or bistable type between 0 and 1, with threshold θ ∈ (0, 1), and "1 more stable than 0" (see Assumption 1.3 below). Then, for any ǫ ∈ (0, 1 − θ) there are two lengths 0 < L ext Among others, the study [3] provides some estimates of the threshold values L ext ǫ and L prop ǫ as ǫ → 0 + , namely As far as we know, for integro-differential equation such as (1.1) there is no such quantitative estimates of this threshold phenomenon in the literature. The goal of the present work is to fill this gap by first deriving sufficient conditions for the existence of similar critical lengths L ext ǫ ad L prop ǫ for suitable levels ǫ > 0, and secondly to provide estimates of these quantities in some asymptotic regime, namely when the height θ + ǫ of the initial data tends to θ.
Note that the analysis in [3] is based on the explicit formula of the heat kernel. For the nonlocal diffusion case, the expression of the "corresponding heat kernel", see (2.2)-(2.3), is much more complicated and estimating its behaviour is far from obvious. However, refined estimates for the tails of the kernel J, as well as for its n−folds convolution J * (n) , can be obtained for a large variety of kernels. Such information, mainly coming from the probability theory, combined with the construction of accurate sub-and super-solutions will allow us to derive some precise estimates of the threshold phenomenon for the nonlocal diffusion equation (1.1).
It turns outs that these estimates of the threshold phenomenon strongly depend on the decay rate at infinity for the kernel J, or tails of J. For instance, when the kernel J has an exponential decay then the critical lengths (when exist) both behave like for the local problem (1.2). On the other hand, for kernels with heavy tails (algebraic decay, Weibull-like tails, etc.), such estimates become more complicated and can take a large variety of different forms.
Let us first present the assumptions on the dispersal kernel J and the nonlinearity f . (i) J is nonnegative, even and satisfies R J(x)dx = 1; (ii) J is nonincreasing in (0, +∞). for some 0 < β ≤ 2 and a > 0.
As observed and proved in [7], expansion (1.3) plays a crucial role in the behavior of the linear equation ∂ t u = J * u−u, but also in some nonlinear phenomena, such as the Fujita blow up phenomenon and the hair trigger effect [1].
Notice that expansion (1.3) contains some information on the tails of J. Indeed, for kernels which have a finite second momentum, namely m 2 (J) := R x 2 J(x)dx < +∞, expansion (1.3) holds true with β = 2, as can be seen in [14, Chapter 2] among others. In particular, this is the case for kernels which are compactly supported, exponentially bounded, or which decrease like O Throughout this note the reaction term arising in (1.1) will satisfy the following set of hypothesis.
and f (u) > 0, ∀u ∈ (θ, 1), and (1.6) In the bistable case, we further require Moreover, in both cases, we require that there are r − > 0 and δ ∈ (θ, 1) such that Notice that for r > 0, the usual cubic bistable nonlinearity satisfies the above assumptions as soon as θ < 1 2 , and so does the ignition nonlinearity The organization of this work is as follows. In section 2, we list some basic facts and key lemmas, in particular some estimates for the tails of n−folds convolution kernel (coming from sum of i.i.d. random variables in probability theory). In section 3 we state our main results including our quantitative estimates on the threshold phenomenon "extinction vs. propagation". In sections 4 and 5 we inquire on extinction and non-extinction phenomena, respectively, in some related toy models. Next, in section 6, we build on these preliminary results to prove our main results. Finally, in section 7 we establish a sufficient condition for "propagation to occur", which is another one of our main results.

Preliminaries
In this section, after presenting some notations and basic facts, we collect some estimates for the tails of J * (i) (i ≥ 1) coming from the probability theory.

Notations and basic linear facts
With this definition, we have, for f , g ∈ L 1 (R), f * g = f g, Formally applying the Fourier transform to equation where ξ is seen as a parameter and which is solved as Applying the inverse Fourier transform, we see that the fundamental solution writes as where δ 0 is the Dirac mass at 0 and J * (i) := J * · · · * J is the convolution of J with itself i − 1 times. Hence the unique bounded solution to ∂ t u = J * u − u with initial data u 0 ∈ L ∞ (R) is given by Obviously, though the convolution of a Dirac mass by a L ∞ function is not pointwise well defined, we let δ 0 * u 0 = u 0 . Moreover, from the normal convergence of the series 2.2 Some key estimates on the tails of J * (i) (i ≥ 1) Observe that if J ∈ L 1 (R) satisfies Assumption 1.2 then so does J * (i) , with i ≥ 1, but with the expansion Notice also that J is a real function since J is even. In the sequel, for i ≥ 1 and L > 0, we denote the tails of J * (i) . The expansion (2.6) on the low frequencies of the Fourier transform J * (i) is very related to the tails of J * (i) . The following first bound is provided by [ As L → +∞, we have the following result quoted from [26,Theorem 5].
(i) If β = 2 then, for any i ≥ 1, one has When 0 < β < 2, the asymptotic behavior of R i (L) is thus very precisely described by Assumption 1.2. On the other hand, when β = 2, Assumption 1.2 is not enough to capture the asymptotic behavior of R i (L): precise estimates, as needed in this work, strongly depend on the decay rate of J at infinity. Below we discuss such estimates for different types of kernels.
The following result deals with exponentially bounded kernels and is known, in probability theory, as the large deviations Cramér theorem (adapted to our situation). We refer the reader to [12, section 2.2] for more details and proofs. The uniformity of the limit (2.12) as stated below is a consequence of [19], see Appendix A. Then, Λ * is nondecreasing on (0, +∞). Also, there exists 0 < L 1 such that Λ * (L) < +∞ for all L ∈ [0, L 1 ] and, for all 0 < L 0 < L 1 , one has Moreover we also have for any L > 0, i ≥ 1 and λ ∈ R, When the kernel J is not exponentially bounded (but still β = 2), there is a large variety of possible tails. The following examples are typical, see e.g. [22], and will be considered in the following.
For these three type classes of kernels, refined estimates of R i (L) when i ≫ 1 and L ≫ 1 are known in the literature: the next lemma is taken from [22, Proposition 3.1].

Main results
For ǫ ∈ (0, 1 − θ] and L > 0, we consider the family of initial data φ ǫ L given by wherein 1 A denotes the characteristic function of the set A. We denote by u ǫ L = u ǫ L (t, x) the solution of (1.1) starting from the initial datum φ ǫ L . The results presented in this section are concerned with the derivation of asymptotic estimates (ǫ ≪ 1) of the size L such that the solution u ǫ L (t, x) goes extinct or propagates at large times. We split our main results into two parts. The first subsection is related to extinction while the second is concerned with propagation.

Extinction results
Let us recall that, for i ≥ 1 and L > 0, R i (L) denotes the tails of J * (i) as defined in (2.7). Our general extinction criterion reads as follows.
Assume T > 0 and L > 0 are such that Then the function u ǫ L is going to extinction at large times, namely The above result has various implications exploring the asymptotic size ensuring the extinction of the solution. More precisely, for each ǫ ∈ (0, 1 − θ], define the interval The next corollaries are concerned with a lower bounded of L ext ǫ as ǫ → 0 + . We start with the case where the kernel J has a slow decay at infinity, meaning that 0 < β < 2 in the expansion (1.3).
Next, we explore a similar question when β = 2. We start with the case of exponentially bounded kernel. Corollary 3.3 (Asymptotic extinction criterion for exponentially bounded kernels) Let Assumptions 1.1 and 1.3 be satisfied and assume further the exponential decay (2.11). Then We pursue the β = 2 case by presenting a series of classical kernels presented in subsection 2.2. (i) Assume that there are c 2 > c 1 > 0, α > 2 and x 0 ∈ R such that c1 for some C − = C − (θ, r + , T, c 1 , α) > 0.

Propagation results
To state our propagation results, we need to assume that for any level θ + ǫ > θ, there exists a size L such that the solution u ǫ L of (1.1) with initial data (3.1) propagates. This assumption will be largely discussed below. We now state our general propagation criterion (let us recall that δ ∈ (θ, 1) is coming from (1.8)).
Theorem 3.6 (Propagation) Let Assumptions 1.1-(i), 1.3 and 3.5 holds. For any α ∈ (0, δ − θ) and m ∈ (0, 1) there exists ǫ 0 > 0 small enough such that, for all ǫ ∈ (0, ǫ 0 ), one has L prop ǫ ≤ L where L satisfies the following condition where we recall that R i (L) denotes the tails of J * (i) as defined in (2.7), In the following we derive an upper bound of L prop ǫ as ǫ → 0 + . As before, this asymptotic strongly depends on the tails of the kernels J * (i) .
We pursue the β = 2 case by presenting a series of classical kernels given in subsection 2.2.

Combining, summarizing and commenting the above results
Let the dispersal kernel satisfy Assumptions 1.1 and 1.2. Let the nonlinearity f be of the bistable or ignition type in the sense of Assumption 1.3. We considered the solution u ǫ L = u ǫ L (t, x) to the one-dimensional reaction integro-differential equation (1.1) starting from the step function φ ǫ L (x) := (θ + ǫ)1 (−L,L) (x). Under the propagation Assumption 3.5, to be discussed in the next subsection, we investigated the values 0 < L ext ǫ ≤ L prop ǫ < +∞ in the asymptotic regime ǫ → 0 + , meaning that the height of the initial step function tends to the threshold value θ of nonlinearity f . For a large variety of dispersal kernels, we proved some estimates stated in subsection 3.1 and 3.2 that we now combine and comment.
In this situation, the "main term" is of magnitude 1 ǫ 1 β , but the lower and upper bounds coincide only up to a logarithmic term.
If m 2 (J) = R x 2 J(x)dx < +∞, meaning β = 2 in Assumption 1.2, we distinguish the following prototype situations under the additional assumption r − ∈ (0, 1), that we believe to be only technical.
If J is exponentially bounded as in (2.11) then In this situation, the lower and upper bounds are sharp in the sense that they coincide up to a multiplicative constant. If J has "not heavy algebraic tails", in the sense that there are c 2 > c 1 > 0, α > 2 and x 0 ∈ R such that c 1 |x| 1+α ≤ J(x) ≤ c 2 |x| 1+α , for all |x| ≥ |x 0 |, and moreover m 2 (J) = 1, then < +∞, (not heavy algebraic tails).
In this situation, the main term is of magnitude 1 ǫ 1 α , but the lower and upper bounds coincide only up to a logarithmic term.
As a conclusion, our analysis reveals that the main term (or, at least, the approximate main term) driving the asymptotics of L ext ǫ and L prop ǫ is, in some sense, selected by the tails of J and can take a large variety of different forms.

Towards Assumption 3.5
The propagation threshold in nonlocal diffusion problems is of fundamental importance and independent interest. In this subsection, we derive some sufficient conditions for the propagation Assumption 3.5 to hold. We place ourselves in the following bistable situation.
Assumption 3.10 (Dispersal kernel and bistable nonlinearity) The dispersal kernel J ∈ C 1 (R) is nonnegative, even, and satisfies R J(x)dx = 1. Assume further J ′ ∈ L 1 (R) and the existence of a finite first moment, namely R |y|J(y)dy < +∞. (3.7) The nonlinearity f ∈ C 1 (R) is bistable in the sense of (1.5)-(1.6) and further satisfies f A traveling wave solution (c, U ) for (1.1) is a speed c ∈ R and a profile U = U (z) solving meaning in particular that u(t, x) := U (x + ct) solves (1.1). The existence, uniqueness and properties of traveling wave solutions in nonlocal diffusion equations with monostable, ignition or bistable nonlinearities has attracted a lot of attention, see e.g. [15], [5], [8], [9,10], [11], [21], [18], [2] and the references therein. Under Assumption 3.10, the existence of traveling waves is studied in the important work of Bates et al. [5] (to which we refer for more precise statements): there exists a monotone solution, in the weak sense, to (3.8). The smoothness of the profile U is a delicate issue. More precisely, assuming further a structure condition on g(u) := u − f (u), see hypothesis (H3) in [5], some of the conclusions of [5, Theorem 3.1] are as follows: if c = 0 then the profile U is smooth and the sign of the speed is that of In particular, a sufficient condition to enforce c > 0 is to assume This shows the relevance of our sufficient conditions for propagation to occur, which we now state.
Theorem 3.11 (Propagation threshold) Let Assumption 3.10 be satisfied and assume 1 0 f (u)du > 0. Assume the existence of (c, U ) an increasing traveling wave solution to (1.1) with c = 0 (and thus c > 0 from the above discussion). Then the conclusions of the propagation Assumption 3.5 hold: for each ǫ ∈ (0, 1 − θ] there exists L > 0 large enough such that u ǫ L (t, x) propagates, in the sense that lim t→+∞ u ǫ L (t, x) = 1 locally uniformly for on R. (3.10) Theorem 3.11 thus provides sufficient conditions for the existence of L prop ǫ < +∞, see (3.5), in the bistable case, and thus in the ignition case due to comparison arguments.
For classical diffusion bistable equations, the fact that "sufficiently large" step initial data lead to propagation is well-known, see [4] for "sufficiently high" step initial data, and [17] for step initial data with any height larger than θ. In the nonlocal diffusion setting, let us mention two propagation threshold properties proved by Berestycki and Rodríguez [6, Theorem 10] and Lim [20, Theorem 1.2] for "sufficiently high" step initial data and under the assumption that the kernel is exponentially bounded in the sense of (2.11). These results also require a structure assumption as (3.9) or a "c > 0 traveling wave" assumption. They mainly rely on the construction of a compactly supported sub-solution using energy methods [6] or traveling wave solutions [20]. Theorem 3.11 improves these results into two directions. First, we cover more dispersal kernels by only requiring the existence of a finite first moment, see (3.7). Second, and even more important, we allow the height of the step initial data which we consider to be θ + ǫ for any small ǫ > 0. This is achieved by performing a nonlocal genralization of the arguments in [17]. This requires a finite first moment of the dispersal kernel to guarantee the existence of a traveling wave solution (c, U ) and to collect some useful integrability properties of the wave U at ±∞. It would be interesting to investigate on the existence (or not) of a propagation threshold for very heavy kernels not admitting a finite first moment, and for step initial data with height θ + ǫ. We believe that this is a delicate issue.
Here r + > 0 and θ > 0 are given and fixed parameter and the subscript + is used to denote the positive part of a real number. We start with a criterion for extinction which does not require ǫ to be small.
Then the solution w = w(t, x) to (4.1) satisfies

3)
and is thus going to extinction at large times.
Proof. Consider v = v(t, x) the solution to the linear Cauchy problem Recalling the definitions of K and ψ in (2.2) and (2.3), one gets v(t, x) = (θ + ǫ)K(t, ·) * 1 (−L,L) (x). Let us first prove, through a regularization argument and [28], that (4.4) Take two sequences (u − n ) n≥0 , (u + n ) n≥0 of continuous, symmetric and nonincreasing on R + functions such that, for all n ≥ 0 and x ∈ R, Now, since the initial data u ± n are continuous, symmetric and decreasing on R + and since the kernel J is symmetric, Xu et al. [28,Theorem 2.5] proved that, for each t > 0 and n ≥ 0, both functions x) are also symmetric and nonincreasing on R + . Hence one obtains, for From (4.5) we deduce that, for any t > 0 and n ≥ 0, Recalling that u ± n (0) = θ + ǫ, we let n → +∞ which, using Lebesgue convergence theorem, yields (4.4). Following Alfaro et al. [3], we aim at constructing a super-solution to (4.1) in the form W (t, x) := v(t, x)ϕ(t), with ϕ(0) = 1 and ϕ(t) > 0. As in [3, Proof of Proposition 2.1], we choose Observe that V (0)ϕ(0) > θ and denote by T > 0 the first time where V (T )ϕ(T ) = θ (obviously we let T = +∞ if such a time does not exist). Then ϕ(t) − θ for all t ∈ [0, T ), and thus one can check that W (t, x) = v(t, x)ϕ(t) is a super-solution to (4.1) on the time interval (0, T ). In particular, if T < +∞, it follows from the comparison principle that w(T, ·) ≤ W (T, ·) ≤ θ, and we are done.
The condition T < +∞ rewrites as: there exists T > 0 such that F L (T ) = 0 where We claim (see below for a proof) that A ′ L (t) ≤ 0 for all t > 0. As a result, since F L (0) = ǫ and Integrating by parts this is equivalent to But, since 0 < A L (s) ≤ 1 (recall (2.5)) and A ′ L (s) ≤ 0 for all s > 0, From this and (4.6), we conclude that (4.2) enforces T < +∞ and is thus a sufficient condition for extinction.
It remains to prove the claim that A ′ L (t) ≤ 0 for all t > 0. Proof. Observe that A L (t) = e −t + R ψ(t, x)u 0 (x)dx, where u 0 = 1 (−L,L) , and thus where we have used (2.4) and the symmetry of ψ and where u = u(t, x) is the solution of ∂ t u = J * u − u starting from u 0 . Now, since both J and u 0 are symmetric and nonincreasing on R + , it follows from Xu et al. [28,Theorem 2.5] (and a regularization argument as above) that u(t, x) ≤ u(t, 0) for all t > 0, x ∈ R. Thus we have A ′ L (t) ≤ 0 for all t > 0. We now take a kernel J satisfying Assumptions 1.1 and 1.2 with, in particular, the expansion (1.3) for some 0 < β ≤ 2. We consider the above extinction criterion in the limit ǫ → 0, revealing the role of the dispersal kernel. Notice that the results below connect to the control issue: given a "final time" T > 0, how large can we choose the size of the initial data so that the solution is everywhere smaller than the threshold θ at time T ?
We start with the case 0 < β < 2. Our analysis seems to reveal that the "final time" does not affect the order of magnitude L ∼ ǫ −1/β but plays a role in the value of the "multiplicative constant". Corollary 4.2 (Asymptotic extinction criterion, 0 < β < 2) Let Assumptions 1.1 and 1.2 be satisfied with 0 < β < 2. Let T > 0 be a fixed time. Then, for any 0 < γ < 1, there exists ǫ 0 > 0 small enough such that, for each ǫ ∈ (0, ǫ 0 ) and for each 0 < L < γ θaCT e −r + T 1/β ǫ −1/β , the solution w = w(t, x) to (4.1) is everywhere smaller than θ at time T and is thus going to extinction at large times. Here constants a > 0 and C = C(β) > 0 come from the asymptotic expansion (1.3) and Lemma 2.2-(ii), respectively.
Proof. For a given 0 < γ < 1, we select an integer N large enough so that Next, from (2.10), there is L 0 > 0 large enough such that, for all L ≥ L 0 and 1 ≤ i ≤ N + 1, Hence, if L ≥ L 0 then As a result, to check the extinction criterion (4.2) it is enough to have L ≥ L 0 and ǫ ≤ γ β θT e −r + T aC L β .
On the other hand, the case β = 2 is more tricky as revealed by the following corollaries.
Proof. We shall use the large deviations Cramér theorem, as stated in Lemma 2.3. We use the criterion (4.2) with a single "large" term i = i(ǫ): to prove extinction it is sufficient to obtain θe −(r + +1)T T i i! R i (L) ≥ ǫ or, equivalently, Recalling that L 1 > 0 is defined in Lemma 2.3 let us fix 0 < L 0 < s 0 < L 1 . Then since Λ * is nondecreasing on (0, +∞), one has 0 ≤ Λ * (L 0 ) ≤ Λ * (s 0 ) ≤ Λ * (L 1 ) < +∞, while according to Lemma 2.3 the convergence in (2.12) holds uniformly in any compact subset of (0, L 1 ]. We now choose where C > 0 is some constant to be determined so that (4.7) is satisfied for all ǫ small enough. From Stirling's formula, we see that there is K > 0 such that 1 i ln(i!) ≤ ln i + K for all i ≥ 1. Hence the condition (4.7) is reached as soon as As ǫ → 0, the right hand side tends to −1/C. On the other hand, we write ǫ −1 , the first term in the right-hand converges to 0 as ǫ → 0 due to the uniformity of the convergence in (2.12), while the monotonicity of Λ * yields, for all ǫ small enough, As a consequence we get lim inf As a result it suffices to have −(r + + 1) − K − Λ * (L 1 ) ≥ −1/C, which is achieved by taking C = C(r + , s 0 ) = C(r + , J) > 0 small enough. The extinction for smaller L follows from the comparison principle. (i) Assume that there are c 2 > c 1 > 0, α > 2 and x 0 ∈ R such that c1 |x| 1+α ≤ J(x) ≤ c2 |x| 1+α for all |x| ≥ |x 0 |. Then there is C − = C − (θ, r + , T, c 1 , α) > 0 such that, for each ǫ > 0 and each L satisfying the solution w = w(t, x) to (4.1) is everywhere smaller than θ at time T and is thus going to extinction at large times.
(ii) Assume that J has regularly varying tails RV (α) with α > 2. Let us fix α ′ > α. Then there exists ǫ 0 > 0 small enough such that, for each ǫ ∈ (0, ǫ 0 ) and for each L satisfying the solution w = w(t, x) to (4.1) is everywhere smaller than θ at time T and is thus going to extinction at large times.
(iii) Assume that J has lognormal-type tails LN (γ, λ, ρ) with γ > 1, λ > 0 and ρ ∈ R. Let us fix λ ′ > λ. Then there exists ǫ 0 > 0 small enough such that, for each ǫ ∈ (0, ǫ 0 ) and for each L satisfying the solution w = w(t, x) to (4.1) is everywhere smaller than θ at time T and is thus going to extinction at large times.
(iv) Assume that J has Weibull-like tails W E(α, λ, ρ) with 0 < α < 1, λ > 0 and ρ ∈ R. Let us fix λ ′ > λ. Then there exists ǫ 0 > 0 small enough such that, for each ǫ ∈ (0, ǫ 0 ) and for each L satisfying the solution w = w(t, x) to (4.1) is everywhere smaller than θ at time T and is thus going to extinction at large times.
Proof. We use the criterion (4.2) with only the term i = 1: to prove extinction it is sufficient to obtain θe −(r + +1)T T R 1 (L) ≥ ǫ.
In case (i) since R 1 (L) ≥ 2c1 α L −α the conclusion easily follows. In case (ii), from (2.14) the condition is recast where S is a slowly varying function. Letting L = 1 which is true for L ≫ 1 from known properties of slowly varying functions, see e.g. [16, VIII, Lemma 2]. As a result, extinction does occur for 0 < ǫ < ǫ 0 with ǫ 0 > 0 small enough and L = 1 ǫ 1 α ′ . The extinction for smaller L follows from the comparison principle.
which is true for L ≫ 1. As a result, extinction does occur for 0 < ǫ < ǫ 0 with ǫ 0 > 0 small enough and The extinction for smaller L follows from the comparison principle. In case (iv), we recast the condition thanks to (2.16) and use a similar argument.
Remark 4.5 As easily seen from the proof, the estimate in case (ii) can be slightly improved (namely α ′ can be taken equal to α) when the slowly varying function S satisfies S(L) → +∞ as L → +∞. Similarly the estimates in (iii) and (iv) can be slightly improved (namely λ ′ can be taken equal to λ) when ρ > 0. Last, in (iv), when ρ = 0, (4.8) can be replaced by

Non-extinction criterion
In this section, we fix r − > 0, θ ∈ (0, 1), and define the linear functioñ For ǫ > 0 and L > 0, we consider w ǫ L = w ǫ L (t, x) the solution of the Cauchy problem We start with a criterion for non-extinction which does not require ǫ to be small.
and thus We now restrict to x satisfying |x| ≤ mL. Since |y| ≥ L ensures that |x| ≤ mL ≤ m|y| and |x − y| ≥ (1 − m)|y|, we deduce that It follows from this and (5.4) that which concludes the proof of (ii).
Remark 5.2 Note that the right hand side of (5.4) is decreasing with respect to L > 0 from 1 − e −Tǫ when L = 0 to 0 when L → +∞.
Proof. Observe first that the existence of L ǫ directly follows from Remark 5.2. We now turn to the asymptotic of L ǫ as ǫ → 0. From Lemma 2.1 we can write from the mean value theorem. Hence, from (1.3), there is C = C(β, a) > 0 such that, for all ǫ > 0 and all L > 0 large enough, From this and the definition of T ǫ in (5.3), we see that the non-extinction criterion (5.4) is satisfied, for all ǫ > 0 small enough, as soon as L > L ǫ : β for some C + = C + (θ, r − , β, a) > 0, which proves the desired result.
On the other hand, the case β = 2 is more tricky as revealed by the following corollaries. Recall that r − is defined in (1.8) and m 2 (J) = R x 2 J(x)dx.
Corollary 5.6 (Asymptotic non-extinction criterion, Weibull-like tails W E(α, λ, ρ = 0)) Let Assumption 1.1-(i) be satisfied. Further assume that J has Weibull-like tails W E(α, λ, ρ = 0) and r − ∈ (0, 1). Let η ∈ (θ, 1) and m ∈ (0, 1) be given. Then for all ǫ > 0, there is L ǫ > 0 such that the conclusion (5.5) holds for all L ≥ L ǫ . Furthermore, there is ǫ 0 > 0 small enough such that, for all 0 < ǫ < ǫ 0 , L ǫ can be chosen as follows, Again the existence of L ǫ in Corollaries 5.4, 5.5 and 5.6 directly follows from Remark 5.2. We now turn to the estimates in Corollaries 5.4, 5.5 and 5.6. We assume As a preparation we consider the sum of the series in (5.4) and roughly show that, for t large enough, the main contribution corresponds to the indexes i around t. To see this, observe that for any t > 0 and any M (t) ∈ N such that M (t) ≤ t one has, using i! ≥ i i e −i and since i → i + i ln t − i ln i is increasing on (0, t), .

(5.8)
On the other hand, for any t > 0 and any N (t) ∈ N such that N (t) > et, one has Let us choose 0 < γ − < 1 such that Then γ − t − 1 < M (t) ≤ γ − t < t and thus (5.8) yields so that Let us next choose N (t) = [3t] + 1 > et, so that (5.9) yields To summarize the above analysis, since J * (i) ≥ 0 and R J * (i) (x)dx = 1 for any i ≥ 1, we have proved the following lemma.
Let us recall that T ǫ = 1 r − ln 2(η−θ) ǫ and define M ǫ := M (T ǫ ) and N ǫ := N (T ǫ ). Then, from Lemma 5.7, we see that the non-extinction criterion (5.4) is satisfied as soon as holds for ǫ > 0 small enough. We now explore the new criterion (5.11) with various assumptions.
As a result the criterion (5.11) is satisfied as soon Since the right hand side tends to 1 − r − > 0 as ǫ → 0, one can choose ℓ > 0 large enough so that the above inequality holds for ǫ > 0 small enough. This completes the proof of Corollary 5.4. Now, we denote d n the threshold sequence as given by Table 1 in Lemma 2.4, and γ n a sequence such that γ n ≫ d n . From the uniform conclusion (2.17) of Lemma 2.4, there is ǫ 0 > 0 such that, for all 0 < ǫ < ǫ 0 , all L ≥ γ Mǫ , As a consequence the criterion (5.11) is asymptotically satisfied as soon as L ≥ L ǫ where where C = C(θ, r − ) is a positive constant and where we recall that . We now rely on Table 1 to compute L ǫ for different kernels in Corollary 5.5.
Proof of Corollary 5.5-(i). In that case, since the second condition in (5.12) is asymptotically satisfied if In view of the line concerning RV (α) in Table 1 of Lemma 2.4, the first condition in (5.12) is also satisfied. This proves Corollary 5.5-(i).
Proof of Corollary 5.5-(ii). For RV (α) with α > 2 and any given 0 <α < α, it follows from (2.14) and known properties of slowly varying functions, see e.g. [16,VIII,Lemma 2], that the second condition in (5.12) is asymptotically satisfied if In view of the line concerning RV (α) in Table 1 of Lemma 2.4, the first condition in (5.12) is also satisfied. This proves Corollary 5.5-(ii).
On the other hand, when 2 3 ≤ α < 1, the first condition in (5.13) prevents such a choice and we are compelled to take , for any given 2 3 ≤ α <α < 1, for the two conditions in (5.13) to be asymptotically satisfied. This proves Corollary 5.5-(iv).
Observe that when 2 3 ≤ α < 1 the above estimate on L ǫ is not so good in particular when α → 1. However for some specific forms of Weibull-like tails W E(α, λ, ρ) with ρ = 0, we can rely on (2.19) to obtain a sharper estimate.
Proof of Corollary 5.6. By a change of variable, the estimate for W E(α, λ, ρ = 0) can be transformed into one of (2.18), namely for W E(α, 1, ρ = 0). From (2.18) and (2.19), we have, up to a multiplicative constant, , one can check that the above right hand side is asymptotically smaller than ǫ 4θ if we choose with C + = C + (J) > 0 large enough. As a result, the non-extinction criterion (5.11) is satisfied. We have thus reached the announced sharper estimate when ρ = 0 (which, when α → 1, is consistent with the exponential case shown in Corollary 5.4). This completes the proof of Corollary 5.6.

Quantitative estimates of the threshold phenomena
In this section, relying on Sections 4 and 5, we complete the proof of the main results of Section 3. We denote u ǫ L = u ǫ L (t, x) the solution to starting from φ ǫ L (x) = (θ + ǫ)1 (−L,L) (x). We start with the extinction results. Proof of Theorem 3.1. By (3.2), u ǫ L is a sub-solution to problem (4.1). As a result, Theorem 3.1 follows from Proposition 4.1 and the comparison principle. Let us now investigate the propagation results.

Propagation threshold
This section is devoted to the proof of the propagation threshold result, namely Theorem 3.11. We shall rely on some ideas developed by Fife and McLeod in [17,Theorem 3.2] and crucially make use of the following integrability properties of the wave profile U . Lemma 7.1 Let Assumption 3.10 be satisfied. Then any monotone traveling wave (c, U ) solving (3.8), and whose existence follows from [5], satisfies the integrability properties Proof. This is nothing else than [5, (5.9)] but, to enlight the importance of the finite first moment hypothesis (3.7) and for the convenience of the reader, we give a proof of the integrability of 1 − U in +∞ (the other one being similar).
From the equation satisfied by U , U (+∞) = 1, and the assumption f ′ (1) < 0, there are R > 0 and C 1 > 0 such that Now we consider ℓ > 0. When c = 0 the regularity of the wave, see subsection 3.4, enables to write thanks to Fubini's theorem. Note that, when c = 0, a mollifying argument as in [2, Lemma 3.2] shows that the above conclusion is still valid. As a result, since 0 ≤ U ≤ 1, we get, for any ℓ > 0, On the other hand, for any ℓ > 0, We thus deduce from (7.1) that, for any ℓ > 0, which implies that 1 − U ∈ L 1 (R, ∞) and completes the proof of the lemma.
We now turn to the proof of Theorem 3.11.
Proof of Theorem 3.11. Let Assumption 3.10 be satisfied and assume 1 0 f (u)du > 0. Consider (c, U ) an increasing traveling wave solution to (1.1) with c > 0. We aim at showing that, for any ǫ ∈ (0, 1 − θ], there exists L > 0 large enough such that propagation occurs for u ǫ L the solution to (1.1) starting from (3.1). To do so and as mentioned above, we rely on the approach of [17] for the classical diffusion case.
Let us consider the function u given by with U ± (t, x) := U (ζ ± (t, x)), where ζ ± (t, x) take the form Here q = q(t) and ξ = ξ(t) are functions to be determined for u to be a sub-solution to (1.1).
From g(0) = 0 and the above claim, g attains its maximum at some t 0 > 0, and For constants ξ 0 < 0 and η 0 > 0 with s 0 := −η 0 − ξ 0 > 0 to be be determined below, we select q(t) := q 0 e −µt + bg(t) = q 0 e −µt + b t 0 e −µ(t−s) (1 − U (cs + s 0 ))ds, t ≥ 0. (7.7) We also let ξ(t) = ξ 0 + η(t) where η is to be selected below with the properties In the sequel, we aim at reaching N u(t, x) ≤ 0 for all x ∈ R, t > 0. Since both u(t, ·) and J are symmetric, it is sufficient to work with x ≥ 0. Since U ′ > 0 we have, for all x ≥ 0 and t > 0, Choose s 0 > 0 large enough so that (1 + b/µ)(1 − U (s 0 )) + q 0 ≤ q 1 . As a consequence, for any such choice, one has, for all t ≥ 0 and x ≥ 0, Below we complete the construction of the sub-solution by investigating the sign of N u(t, x) for x ≥ 0 and t > 0. To do so, recalling that δ > 0 was chosen above for (7.3) to hold, we split our analysis according to the value of U − (t, x).
From this we, again, deduce (7.10) and conclude as in the first case.
It follows from the comparison principle that u(t, x) ≥ u(t, x) for all t ≥ 0, x ∈ R. Since u satisfies (3.10), so does u and the proof is complete.
A Appendix: the uniformity in (2.12) In this Appendix, we show that the uniformity of the limit (2.12) is a consequence of an important estimate taken from [19].
Moreover due to (i), one has m(0) = 0. Define also the function a : m(A) → R by a = m −1 , so that 0 ∈ m(A), a(0) = 0 and a is continuous and increasing. Using the above notations, [19, Theorem A (i)] reads as follows. Equipped with this we can conclude on the uniformity of the limit (2.12) as stated in Lemma 2.3.