When fast diffusion and reactive growth both induce accelerating invasions

We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

1. Introduction. In this paper, companion of [3], we are concerned with the spreading properties of u(t, x) the solution of the nonlinear monostable reactiondiffusion equation in the fast diffusion regime 0 < m < 1, the linear diffusion case m = 1 and the porous medium diffusion regime m > 1 being already well understood. The typical nonlinearity f we have in mind is f (s) = rs β (1 − s), with r > 0 and β > 1 (Allee effect), the Fisher-KPP case β = 1 being already well understood. Equation (1) is supplemented with a nonnegative initial data which is front-like and may have a heavy tail, say u 0 (x) 1 x α for some α > 0 as x → +∞ (see Assumption 1.1). Precisely, our goal is to understand the regime In [3], we proved that, in this regime, propagation occurs by accelerating but a precise estimate of the position of the level sets of u(t, ·) as t → +∞ was missing. In the present paper, we fill this gap by constructing very refined sub and supersolutions, which rely on self-similar solutions of In particular, and roughly speaking, we show that the leading term of the position of the level sets is of the monomial type t β−m 2(β−1) , which is independent on α, thus on the tail of the initial data. This is in contrast with the other regimes fully described in [3].
We refer to the introduction in [3] for references, comments and relevance in population dynamics models on the three main effects inserted in the Cauchy problem (1): nonlinear diffusion [15,17,4], Allee effect [6] (vs KPP nonlinearities [7,16]), tail of the initial data. Let us briefly recall the available results on the propagation of solutions to (1), with a front-like initial data (whose behaviour at +∞ is of crucial importance on the speed of invasion).
• In presence of an Allee effect (β > 1): the linear diffusion (m = 1) case was studied in [1]. For algebraic tails, the exact separation between acceleration or not (depending on the strength of the Allee effect) was obtained: when β < 1 + 1 α acceleration occurs, and the location of the level sets of the solution is of the monomial type t 1 α(β−1) as t → +∞. The nonlinear diffusion cases were recently studied in [3]: in the porous medium diffusion case m > 1, the obtained results were sharp and very similar to the case m = 1. On the other hand, because of the possible acceleration induced by fast diffusion itself, the case 0 < m < 1 is much more subtle. We proved in [3] that acceleration occurs if and only if β < max 1 + 1 α , 2 − m . Next, in the range β < min 1 + 1 α , m + 2 α , we precisely estimated the position of the level sets, again of the monomial type t 1 α(β−1) . The keystone for constructing accurate sub and supersolutions was the solution w(t, x) of the ODE Cauchy problem (x playing the role of a parameter) On the other hand, in the remaining parameter range, which rewrites as (2), the acceleration is induced by a combination of fast diffusion and of reactive growth. Since (4) neglects the former, it was not enough to precisely quantify the acceleration phenomena. The main novelty of the present paper is to use a self-similar solution of (3) to build accurate sub and supersolutions, which enable to understand the acceleration regime (2) that was missing in [3]. Through this work, we make the following assumption on the initial condition.
Assumption 1.1 (Initial condition). The initial condition u 0 : R → [0, 1] is uniformly continuous and asymptotically front-like, in the sense that and for some α > 0, C > 0 and x 0 > 1. and In the sequel, we always denote by u(t, x) the solution of (1) with initial condition u 0 . From the above assumptions and the comparison principle, one gets 0 ≤ u(t, x) ≤ 1 and even Also, since the initial data is front-like, the state u ≡ 1 does invade the whole line meaning that propagation is at least linear. We also have lim x→+∞ u(t, x) = 0, ∀t ≥ 0.
If we assume furthermore that α ≥ 2 1−m then the upper bound is sharply improved to x + (t) : where z 0 > 0 is a constant depending on m, β and r (but not on ε), and not on α nor C (the initial data is irrelevant for the upper bound).
Remark 1. As a matter of fact, the upper estimate (13) can be extended (up to changing the multiplicative constant) to the regime The proof is rather similar and simply relies on a different choice of a selfsimilar solution with a slower decay at infinity. To make the presentation simpler, we treat this case separately and only sketch the necessary changes in Remarks 2 and 4. We point out that the multiplicative constant in (13) may still be chosen independently of the initial data as long as m + 2 α < β. The paper is organized as follows. In Section 2, we state the existence of an adequate self-similar solution of equation (3), which is the main tool for construction of sharp sub and supersolutions. The latter is achieved in Section 3, thus proving Theorem 1.3. Last, the actual construction of the self-similar solution is performed in Section 4.

2.
A key self-similar solution. Guided by [15], we plug the self-similar ansatz into and obtain after some straightforward computations that one needs In this section, we claim the existence of such a self-similar solution having the required asymptotics properties for our analysis to work in Section 3. The proofs are postponed to Section 4. In the range (10) of Theorem 1.3, there are z 0 > 0, C 0 > 0, C ∞ > 0 such that the following holds: for any k 0 < C 0 < K 0 , any 0 < k ∞ < C ∞ and some K ∞ > C ∞ , there is a decreasing function ϕ : (z 0 , +∞) → (0, +∞) solving (16) on (z 0 , +∞) and satisfying the following boundary estimates k 0 , in a right neighborhood of z 0 , say (z 0 , z 1 ), (17) and in a neighborhood of +∞.
Hence we are equipped with (14) solving (15) in the domain t > 0, x > z 0 t β−m 2(β−1) . For later use, we use the convention The blow-up zone. We will also need the following estimates: up to reducing z 1 , we have that where s 0 > 0 is as in (6). We also select δ > 0 small enough so that We define as the self-similar solution (14) of Section 2, solving (15) with r replaced by r − ε. Moreover, we choose the constants k 0 and K 0 in (17) close enough to C 0 so that (21) holds, and so do all estimates of Section 2. Now, for any t > 0, from ϕ(z 0 ) = +∞, ϕ(+∞) = 0 and the monotonicity of ϕ, we can define Notice that from expression (32) we clearly have X(t) < z 1 t β−m 2(β−1) for t > 0 large enough.
Then there is T > 0 large enough so that v(t, x) is a subsolution to (1) in the domain (T, +∞) × R.
Proof. Let us note that v is smooth in both subdomains {x < X(t)} and {x > X(t)}. Also, it is continuous in (0, +∞) × R as well as C 1 with respect to x at the junction point X(t). This means that a comparison principle is applicable provided that v satisfies is obviously a subsolution to (1), we only need to check this inequality when t > T , x > X(t). First it is straightforward that v(t, x) ≤ max w≥0 w(1 − Aw η ) = 1 (A(1+η)) 1/η η 1+η ≤ s 0 in view of (30). It then follows from (6) and a convexity inequality that Next we have Also, we have Plugging (35), (36) and (37) into (34) we arrive at The second term in the above right hand side member is nonpositive in view of (29). We deduce from (33) Now we need to estimate the derivatives of w by powers of w. To do so, we distinguish two regions. The far away zone t > 0, x ≥ z 1 t β−m 2(β−1) , where we take advantage of (27) and (28) to infer from (38) that Hence, for a positive constant K = K(η, m, γ), we have , that is after a large time and in the far away zone.
, where we take advantage of (22), (23) and (24) to infer from (38) that Hence, for a positive constant still denoted K = K(η, m, γ), we have , that is after a large time and in the blow-up zone.
Proof. By the comparison principle, we can assume without loss of generality that u 0 is nonincreasing on R, and therefore, from the comparison principle again, Moreover, we have from (27) that Recall also that, up to enlarging Next, from the invasion result (8), there is T 0 > T such that Last, by comparison with the fast diffusion equation -namely ∂ t u ≥ ∂ xx (u m )and thanks to [14,Theorem 2.4], we know that for any for x large enough, and moreover C(T ) → +∞ as T → +∞. Hence, we can now fix T > T 0 so that, for some X 2 = X 2 (T ) > X 1 , Now we define X := X 2 − X 1 and prove (39), by dividing into three regions. When x ≤ X 2 , so that x − X ≤ X 1 , this follows from (43) and the upper line in (42). When X 2 ≤ x ≤ 2X 2 − X 1 , so that X 1 ≤ x − X ≤ X 2 , we successively use (40), (44), the second line in (42) and (41) to get When x ≥ 2X 2 − X 1 , so that x − X ≥ X 2 , we successively use (44) and the second line in (42) to obtain This completes the proof of (39).
From Lemma 3.1 and Lemma 3.2, we deduce that Now, the proof is the same as that in [1] or [3, subsections 5.1 and 6.2]. Roughly speaking, the subsolution "lifts" the solution u(t, x) on intervals that enlarge with the correct acceleration, which provides the lower bound in (12) on the level sets E λ (t) when λ is small. Next, the estimate for larger λ is obtained thanks to the fact that invasion occurs for front-like initial data, see (8). We omit the details and conclude that the lower bound in (12) is proved.

3.2.
Upper bound on the level sets in (11). Let λ ∈ (0, 1) and ε > 0 small be given. The expression of x + (t) in (12) was already proved in [3]. We thus assume α ≥ 2 1−m and look after the improvement (13). Again the self similar solution of Section 2 provides a more accurate supersolution.
In view of α ≥ 2 1−m , the upper bound in (5), and the comparison principle, it is enough (to prove the upper estimate on the level sets) to consider the case where We define w(t, x) as the self-similar solution (14) of Section 2, solving (15) with r replaced by r (so that z 0 , C ∞ ... are replaced by z 0 , C ∞ ...). From (7) it is immediate that is a supersolution for equation (1). Recalling convention (19), notice that ψ(t, , which enforces x > x 0 in view of the second inequality in (46), this is a consequence of (45) and the third inequality in (46); last, in the range x ≥ z 1 T β−m 2(β−1) this follows from (25), the first inequality in (46), and (45).
Hence, it follows from the comparison principle that For t ≥ t λ and x ∈ E λ (t), it follows that w(t + T, x) ≥ λ which, using the expression for w transfers into .

MATTHIEU ALFARO AND THOMAS GILETTI
From the properties of ϕ we infer that → z 0 as t → +∞ and thus, from (17), for t large enough. Hence for t ≥ T λ,ε chosen sufficiently large. This concludes the proof of the upper bound in (13).

Remark 2.
We point out that (32) provides a supersolution whatever the choice of a self-similar solution ϕ. In particular, all the asymptotics which we established in Section 2 are necessary only for the construction of the subsolution, which as usual is the more intricate part of the proof. Therefore in the regime m + 2 α ≤ β < 2 − m, 1 1−m < α < 2 1−m , replacing ϕ by ϕ a solution of (16) blowing up at some point z 0 and decaying at infinity with asymptotics one can repeat a similar argument to the above to find an upper estimate on the position of any level set in the form of (13); this was announced in Remark 1. We also refer to Remark 4 below for the existence of such a function ϕ.

4.
Actual construction of the self-similar solution. As far as the construction of self-similar solutions is concerned, let us mention the strategy of [10] -see also [13,11] for related results -which mainly consists in using an integral formulation of the problem. Because of a non integrable singularity in the problem under investigation, this seems quite impracticable and we therefore adopt a different approach through sub and supersolutions. Another difficulty is that we are looking for both limiting behaviours to be, in some sense, critical, and therefore we have to "shoot" simultaneously in both directions. To do so, we start by looking for a solution that has the appropriate blow-up profile, with the blow-up point z 0 being any positive number. We will then show that if z 0 is too large, this blow-up solution decays "slowly" at infinity, while if z 0 is small it decays "quickly". This will lead us to find a particular z 0 where the solution has both wanted asymptotics at the blow-up point and at infinity. 4.1. Comparison principles. As mentioned above, we will use sub and supersolutions to construct the self-similar solution of Section 2. We state here some properties that we will use extensively. First, we say that ψ is a subsolution of (16) if it satisfies the differential inequality If moreover ψ is not a solution, then we call it a strict subsolution. Similarly, we define the notion of supersolution and strict supersolution. Due to the singularity of the equation as ψ → 0, we will only consider positive (sub and super) solutions.
Let us already point out that any "nontrivial shift to the left" of a decreasing subsolution is a strict subsolution, whereas any "nontrivial shift to the right" of a decreasing supersolution is a strict supersolution. This follows from a straightforward computation. For later use, we state this in the following proposition.
Our main tool will be a comparison principle, which we establish through a sliding argument and thanks to the previous proposition.
Proposition 2 (A comparison principle). Let ψ 1 and ψ 2 be respectively a sub and a supersolution of (16) on an open interval I. Furthermore, we assume that both functions are decreasing.
We will also need some small variations of the above comparison principle. Let us first highlight the useful following extension of Proposition 2.
Next, we also point out that, in the sequel, we will occasionally use slightly different sliding arguments to reach a similar comparison property in some situations where the functions ψ 1 and ψ 2 are not necessarily decreasing. Finally, this comparison principle can also be extended to the strict ordering of blow-up points.
Proposition 4 (Ordering blow-up points). Let ψ 1 and ψ 2 be respectively a sub and a supersolution of (16). Furthermore, we assume that both functions are decreasing, and that ψ 2 blows up at some point Z 2 .
Proof. By the previous proposition and the subsequent remark, we already know that ψ 1 ≥ ψ 2 on the left of z 1 (and in the intersection of both their intervals of definition). In particular ψ 1 blows up at some Z 1 ≥ Z 2 and it only remains to show that ψ 1 ≡ ψ 2 implies that Z 1 > Z 2 . We proceed by contradiction and assume that there exists some pointz on the left of z 1 where ψ 1 (z) > ψ 2 (z), yet Z 1 = Z 2 . For any a > 0, the shifted to the left ψ 1 (·+a) subsolution is (strictly) smaller than ψ 2 in neighborhoods of Z 2 (thanks to Z 1 = Z 2 ) and of z 1 − a (thanks to the monotonicity assumption). Moreover, if a is small enough, then ψ 1 (z + a) − ψ 2 (z) has to be positive, so that both functions intersect at least twice. Using Proposition 3, we see that ψ 1 (· + a) ≥ ψ 2 on a neighborhood of Z 2 , contradicting the fact that it blows up at Z 2 − a.
We point out that an immediate corollary of Proposition 4 is that two different solutions blowing up at the same point cannot intersect.

4.2.
Existence of a solution with the appropriate blow-up profile. In this subsection, we fix z 0 any positive real number. Our construction relies on several sub and supersolutions which we detail below. For C > 0, we let which we aim at plugging in equation (16). We compute Thanks to the fact that m + 2β − 2 < β, we find that there is some such that, for any C > C 0 , ψ 0,C satisfies in a neighborhood of z 0 (strict subsolution) while, for any C < C 0 , ψ 0,C satisfies the opposite inequality (strict supersolution). Let us be more specific concerning the neighborhoods of z 0 . For any γ > 1, let k 0 := C0 γ and K 0 := γC 0 . Then ψ 0,K0 is a subsolution of the ODE (16) if Since the second term in the left hand side (diffusion term) and the right hand side are positive, a sufficient condition is given by where C 1 is positive. Notice that the right hand side does not depend on z 0 . Notice also that, recalling the expression of C 0 above, we can rewrite this as One can also make more precise the neighborhood where ψ 0,k0 is a supersolution.
We recall that, according to Proposition 1, any shift ofψ or ψ 0,K0 to the left remains a subsolution, while any shift to the right of ψ 0,k0 remains a supersolution.
We first prove that neither set is empty. Indeed, in view of the ODE (16) a solution cannot reach a positive minimum and thus any ξ < 0 enforces the solution ϕ θ,ξ to crossψ, so that (−∞, 0) ⊂ Ξ 1 . On the other hand a direct computation shows that, if the slope p > 0 is large enough, then the linear function w(z) := −p(z − (z 0 + δ(z 0 ))) + θ is a supersolution on an interval (z p , z 0 + δ(z 0 )) where it crosses ψ 0,k0 exactly once. Now choose ξ > p, and let us prove that ϕ θ,ξ also crosses ψ 0,k0 in the same interval. To do so, we use a sliding argument: notice that the solution ϕ θ,ξ may not be decreasing, so that Proposition 2 does not apply directly. Nonetheless, we proceed by contradiction and assume that ϕ θ,ξ ≤ ψ 0,k0 in (z p , z 0 + δ(z 0 )). In particular, the supremum of ϕ θ,ξ in the same interval is less than w(z p ). One can then reproduce the exact same argument as in the proof of Proposition 2 to obtain a critical shift a 0 such that w(· − a 0 ) − ϕ θ,ξ admits a zero minimum in the interval (z p + a 0 , z 0 + δ(z 0 )). This contradicts the differential inequality and equality satisfied by both functions w and ϕ θ,ξ . It follows that Ξ 2 is not empty either.
By continuity of the solutions of the ODE with respect to the slope parameter ξ, Ξ 1 and Ξ 2 are open sets.
Also, by a comparison argument, if ξ ∈ Ξ 1 then (−∞, ξ] ⊂ Ξ 1 . Indeed, choose any ξ < ξ and assume by contradiction that ϕ θ,ξ ≥ψ on the left of z 0 . In particular, ϕ θ,ξ blows up at some point z ξ ≥ z 0 and, since as we explained above it cannot reach a positive minimum, it has to be decreasing on its interval of definition. We again use a sliding argument and find some a 0 > 0 so that the function ϕ θ,ξ (· − a 0 ) − ϕ θ,ξ reaches a zero minimum inside the interval (z ξ + a 0 , z 0 + δ(z 0 )). This gives a contradiction and we conclude that ξ ∈ Ξ 1 .
Hence, thanks to the monotonicity of ψ 0,k0 and the fact that ψ 0,k0 (z) → +∞ as z → z 0 , we conclude that ψ 0,k0 (· − a 0 ) − ϕ θ,ξ reaches a zero minimum value at some point, which is contradicted by the differential inequality for the supersolution ψ 0,k0 (· − a 0 ) and the ODE for the solution ϕ θ,ξ . Therefore, it remains to rule out the case when ϕ θ,ξ only crosses ψ 0,k0 once, andψ twice. Recalling that ϕ θ,ξ cannot change monotonicity more than once, it then follows that it is a decreasing function. A straightforward use of Proposition 2 (which is now applicable) contradicts the fact that ψ 0,k0 crossesψ twice. Finally ξ 1 ≤ ξ 2 and the claim follows by taking any ξ ∈ [ξ 1 , ξ 2 ].
The set of such θ is not empty by the above claim (recall that ψ 0,k0 < ψ 0,K0 ), and obviously it is bounded from above as the inequality fails when θ ≥ ψ 0,K0 (z 0 +δ(z 0 )). It follows that θ * is well-defined and finite. Now take sequences θ n θ * and ξ n , where ξ n is chosen so that ϕ θn,ξn blows up at z 0 and lies below ψ 0,K0 .
Claim 4.2. The sequence ξ n is bounded. Hence, up to extraction of a subsequence, ξ n → ξ * .
Proof. If ξ n < 0 the solution ϕ θn,ξn , which blows up at z 0 , has to reach a positive minimum at some point, where we test the equation to reach a contradiction. Hence ξ n ≥ 0 and moreover ϕ θn,ξn is a decreasing function. Using yet another comparison argument, we obtain that for any ξ ≤ ξ n the solution ϕ θn,ξ lies below ψ 0,K0 : if not and using again the fact that solutions cannot reach a positive minimum at any point, then ϕ θn,ξ has to be decreasing in some left interval of z 0 + δ(z 0 ) where it crosses ψ 0,K0 , and applying Proposition 2 to the pair of functions ϕ θn,ξn and ϕ θn,ξ one immediately reaches a contradiction. Now proceed by contradiction and assume that ξ n → +∞ (even up to extraction of a subsequence). Then by a limiting argument we get that for any ξ ∈ R, the solution ϕ θ * ,ξ lies below ψ 0,K0 , which is impossible (see the non emptiness of Ξ 2 in the proof of Claim 4.1). The claim is proved.
Let us check that ϕ θ * ,ξ * blows up at z 0 . Clearly the above inequality implies that it cannot blow up on the right of z 0 . Now proceed by contradiction and assume that ϕ θ * ,ξ * is finite at z 0 . Using again the continuity of solutions of the ODE, we have a small open neighborhood of (θ * , ξ * ) such that the solution is again finite at z 0 : this is a clear contradiction with our construction. We conclude, as announced, that ϕ θ * ,ξ * blows up at z 0 .
As we have pointed out several times, if z * is a critical point of any (positive) solution, the equation (16) yields that z * is a strict local maximum point. In particular, blow-up cannot occur on the left of z * , and hence the following holds. Before we proceed, let us extend the previous upper inequality: we show that where κ comes from (48) and is such that ψ 0,K0 is still a subsolution in this enlarged interval (z 0 , z 0 + κz 0 ). Assume by contradiction that there is a contact point z 1 ∈ (z 0 , z 0 + κz 0 ) and without loss of generality that ϕ θ * ,ξ * ≤ ψ 0,K0 in (z 0 , z 1 ). Since we are equipped with a strict subsolution and a solution we also have that ϕ θ * ,ξ * ≡ ψ 0,K0 on (z 0 , z 1 ). Proposition 4 immediately contradicts the fact that both functions blow up at the same point z 0 . Therefore (49) holds.
We proceed again by contradiction and assume that ϕ θ * ,ξ * (z 0 +δ ) < ψ 0,k0 (z 0 +δ ) for some δ ∈ (0, δ(z 0 )). The idea is to show that ϕ θ * ,ξ * falls into the value range (betweenψ and ψ 0,k0 ) where blow-up is expected with "slow" asymptotics, as the solutions we have constructed in Claim 4.1. Therefore, small perturbations should also be in the same value range, leading to a contradiction with the "critical" choice of θ * .
We again used Proposition 4, and a strong maximum principle for this inequality to be strict. Now considerφ the solution of the ODE (16) with initial conditionŝ Here we chooseθ > ϕ θ * ,ξ * (z 0 + δ ) but very close, and by a similar argument as that of Claim 4.1, we can find aξ such thatφ remains betweenψ and ψ 0,k0 in the interval (z 0 , z 0 + δ ). Since both solutions blow up at the same point and by Proposition 4, we have thatφ and ϕ θ * ,ξ * cannot intersect. In particular, noting thatφ is also defined on the right of z 0 + δ , we get that θ * = ϕ θ * ,ξ * (z 0 + δ(z 0 )) <φ(z 0 + δ(z 0 )).
We sum up our result in the next proposition.
Proposition 5 (Blow-up solutions). For any z 0 > 0, any k 0 < C 0 < K 0 , there exists a decreasing solution ϕ z0 of the ODE (16) such that in some right neighborhood of z 0 .
Furthermore, when γk 0 = K0 γ = C 0 with γ > 1, the upper inequality holds true on an explicit interval, see (48) and (49). The lower inequality holds in an interval (z 0 , z 0 + δ(z 0 )), see (50), where δ(z 0 ) can be chosen to be continuous with respect to z 0 . Remark 3. In particular, the above statement implies that, given a sequence z n → z 0 , the associated solutions ϕ zn satisfy estimates on intervals that "do not disappear" when passing to the limit. In particular, any limit of the solutions ϕ zn (which exists by usual estimates, up to extraction of a subsequence) has the wanted blow-up behaviour too.

4.3.
Behaviour of ϕ z0 at infinity. We will next investigate the behaviour of ϕ z0 as z → +∞, depending on the choice of the blow-up point z 0 . Let us first introduce some sub and supersolutions decaying to 0.
For any C > 0, we define A straightforward computation shows that there is C ∞ = C ∞ (m) > 0 so that ψ ∞,C∞ satisfies the ODE (16) without the reaction term (which is negligible at infinity), that is (16) with r = 0. In particular, ψ ∞,C∞ satisfies the differential inequality