Non-local competition slows down front acceleration during dispersal evolution

We investigate the super-linear spreading in a reaction-diffusion model analogous to the Fisher-KPP equation, but in which the population is heterogeneous with respect to the dispersal ability of individuals, and the saturation factor is non-local with respect to one variable. We prove that the rate of acceleration is slower than the rate of acceleration predicted by the linear problem, that is, without saturation. This hindering phenomenon is the consequence of a subtle interplay between the non-local saturation and the non-trivial dynamics of some particular curves that carry the mass at the front. A careful analysis of these trajectories allows us to identify the value of the rate of acceleration. The article is complemented with numerical simulations that illustrate some behavior of the model that is beyond our analysis.


E N C A S D ' É V O L U TIO N D E L A DIS P E R SIO N
Abstract. -We investigate super-linear spreading in a reaction-diffusion model analogous to the Fisher-KPP equation, but in which the population is heterogeneous with respect to the dispersal ability of individuals and the saturation factor is non-local with respect to one variable. It was previously shown that the population expands as O(t 3/2 ). We identify a constant α * , and show that, in a weak sense, the front is located at α * t 3/2 . Surprisingly, α * is smaller than the prefactor predicted by the linear problem (that is, without saturation) and analogous problem with local saturation. This hindering phenomenon is the consequence of a subtle interplay between the non-local saturation and the non-trivial dynamics of some particular curves that carry the mass to the front. A careful analysis of these trajectories allows us to characterize the value α * . The article is complemented with numerical simulations that illustrate some behavior of the model that is beyond our analysis.

Introduction and Main result
It is commonly acknowledged that the rate of front propagation for logistic reactiondiffusion equations is determined by the linear problem, that is, without growth saturation. This is indeed the case for the celebrated Fisher-KPP equation, It is known [AW78,Fis37,KPP37] that the level lines of the solution propagate asymptotically with speed 2 √ θ, provided the initial data is localized (e.g., compactly supported). This coincides with the spreading speed of the linear problemn t = θn xx +n, which can be seen, for instance, from its fundamental solution, The linear determinacy of the wave speed for reaction-diffusion equations is a longstanding question, see e.g. [Cro03,HR75] and [LMN04,Wei12] for recent developments on scalar homogeneous equations. It has been established in many other contexts, such as for related inhomogeneous models (see, e.g., [BH02,BHN05], and the recent [NR17] and references therein) as well as for systems under certain conditions (see, e.g. [LLW02,LWL05,WLL02], the recent work in [Gir18a,Gir18b], and references therein). More recently, linear determinacy has been established for many non-local equations as well (see, e.g., [ACR13,BJS16,BNPR09,HR14]). This is necessarily only a small sampling of the enormous body of literature utilizing the relationship between spreading speeds and linearization in reaction-diffusion equations arising in ecology and evolution.
In the present work, we report on a similar equation, called the cane toads equation, that describes a population that is heterogeneous with respect to its dispersal ability. Namely, we consider the population density f (t, x, θ) whose dynamics are described by the following equation: where ρ(t, x) = ∞ 1 f (t, x, θ)dθ is the spatial density. The zeroth order term f (1 − ρ) is referred to as the reaction term. The equation is complemented with a measurable initial datum f 0 such that, for some θ in , C 0 1, up to a set of measure zero.
Equation (1.2) was proposed as a minimal model to describe the interplay between ecological processes (population growth and migration) and evolutionary processes (here, dispersal evolution) during the invasion of an alien species in a new environment, see [BCMV12] following earlier work in [CM07] and [PDA11]. The population is structured with respect to the dispersal ability of individuals, which is encoded in the trait θ > 1. Offspring may differ from their parents with respect to mobility. Deviation of mobility at birth is accounted for as f + f θθ , with Neumann boundary conditions at θ = 1. Finally, growth becomes saturated as the population density ρ(t, x) reaches unit capacity locally in space. We note that we use the trait space θ ∈ (1, ∞) for simplicity, but our proof applies to the case when the trait space is (θ, ∞) for any θ > 0.
Problem (1.2) shares some similarities with kinetic equations (see, for example, the review [Vil02]), as the structure variable θ acts on the higher order differential operator. However, here the differential operator is of second order, whereas it is of first order (transport) in the case of kinetic equations.
The goal of this study is to understand spreading in (1.2), and, in particular, to emphasize the comparison with the rate of propagation of the linearized problemf t = θf xx +f θθ +f . Indeed, our main results, Theorems 1.2 and 1.3, imply that propagation in (1.2) is slower than that predicted by the linear problem. This is surprising at first glance, as, to our knowledge, there are no settings in which linearization in homogeneous scalar reaction-diffusion equations overestimates the asymptotic wave speed [Cro03,HR75,LMN04], whereas it does overestimate the spreading rate in our case study. TOME 5 (2022) is located, roughly speaking, between (8/3 7/4 )t 3/2 and (4/3)t 3/2 in a weak sense (see [BHR17b, Theorem 1.2]).
Here, we establish that, contrary to previous predictions [BCM + 12], the spreading of the population is, in a weak sense, slower than (4/3)t 3/2 for (1.2). Namely, there exists a constant α * ∈ (0, 4/3) such that the "front" is located around α * t 3/2 , see Theorem 1.2 for a precise statement. By refining some calculations performed in [BHR17b], we can prove without too much effort that α * > 5/4 > 8/3 7/4 . Characterizing α * requires more work. We find eventually that α * is the root of an algebraic equation involving the Airy function and its first derivative. This allows to get a numerical value for α * of arbitrary precision, e.g. α * ≈ 1.315135. It is immediate to check that this value is compatible with all previous bounds. Indeed, we notice that α * is much closer to 4/3 than any of the above lower bounds, so that the relative difference is below 2%.

Abstract characterization of the critical value α *
In order to give precise results, we need some notation. Let α ∈ [0, 4/3] and µ 1/2. Let U α,µ denote the value function of the following variational problem: (1.4)
We have the following properties of α * : Proposition 1.1. -The constant α * is well-defined and it satisfies 5/4 < α * < 4/3. We refer to Section 3.1 and Section 4 for the proof of this statement.
Interestingly, the value α * gives a reasonable (but weak) description of the spreading properties of (1.2).
Theorem 1.2. -Suppose that f satisfies (1.2) with initial data f 0 localized in the sense of (1.3). Then, Moreover, for all > 0 there exists δ > 0 such that Roughly, following Theorem 1.2, there exist infinite sequences of times t 1 , t 2 , . . . → ∞ and s 1 , s 2 , . . . → ∞ such that the level line {ρ = δ} has reached α * t 3/2 i at t i , whereas the level line {ρ = 1/2} is no further than α * s 3/2 i at s i . A few comments are in order. First, as with other non-local Fisher-KPP-type equations that lack the comparison principle [BNPR09], we are unable to establish spatial monotonicity of ρ and f . Thus, we cannot rule out that the front oscillates, in contrast to what is depicted in Figure 1.1. Second, we cannot rule out front stretching, even along sequences of times (this is the situation depicted in Figure 1.1). The reason is that the upper threshold value 1/2 cannot be made arbitrarily small in our approach. Our result is compatible with a monotonic front in which X 1/2 (t) is moving at rate (5/4)t 3/2 and X 1/10 (t) is moving at rate (4/3)t 3/2 , for instance. Getting stronger results following our approach seems out of reach at present.
The appendix is devoted to numerical computations that indicate that the front profile is monotonically decreasing and that all level lines move at the same rate. Together with Theorem 1.2, this suggests that the front propagates at rate α * t 3/2 in the usual sense.

Further characterization of the critical value α *
We give two other characterizations of α * . The following definitions are required. Let Ξ 0 ≈ −2.34 be the largest zero of the Airy function Ai. For ξ > Ξ 0 , we define the function Note that Ξ 0 is a singular point for R, and that R is well-defined and smooth on (Ξ 0 , ∞). We provide further discussion of these and related functions in Section 7.3. In addition, we define the following algebraic function V for τ ∈ (0, 1), The constant α * has the following two characterizations.
(i) For all α ∈ (0, 4/3], we have Hence, such that the argument of R belongs to (Ξ 0 , ∞). Then, The main purpose of Theorem 1.3 is to provide an analytic formula for α * that can be easily (numerically) computed. It is from this representation that we obtain the decimal approximation α * ≈ 1.315135 given above.
We mention that, in fact, a stronger scaling relationship than (1.10) holds that takes into account the scaling in θ. This is straightforward to obtain and Theorem 1.3 (i) follows directly from it. The advantages of Theorem 1.3 (i) are twofold. First, it provides a direct way of determining that α * < 4/3. Second, it reduces the task of finding α * to computing only U 4/3 , instead of the whole range of U α for α ∈ [0, 4/3]. Theorem 1.3 (i) also allows us to simplify the proof of Theorem 1.3 (ii).

Motivation and state of the art
The interplay between evolutionary processes and spatial heterogeneities has a long history in theoretical biology [CBBB12]. It is commonly accepted that migration of individuals can dramatically alter the local adaptation of species when colonizing new environments [MW01]. This phenomenon is of particular importance at the margin of a species' range where individuals experience very low competition with conspecifics, and where gene flow plays a key role. An important related issue in evolutionary biology is dispersal evolution, see e.g. [Ron07].
An evolutionary spatial sorting process has been described in [SBP11]. Intuitively, individuals with higher dispersal reach new areas first, and there they produce offspring having possibly higher abilities. Based on numerical simulations of an individual-based model, it has been predicted that this process generates biased phenotypic distributions towards high dispersive phenotypes at the expanding margin [TD02,TMBD09]. As a by-product, the front accelerates, at least transiently before the process stabilizes. Evidence of biased distributions and accelerating fronts have been reported [PBWS06, TBW + 01]. It is worth noticing that ecological (species invasion) and evolutionary processes (dynamics of phenotypic distribution) can arise over similar time scales.
Equation (1.2) was introduced in [BCMV12] and built off the previous contributions [CM07] and [PDA11]. It has proven amenable to mathematical analysis. In the case of bounded dispersal (θ ∈ (1, 10), say) Bouin and Calvez [BC14] constructed travelling waves which are stationary solutions in the moving frame x − c * t for a well-chosen (linearly determined) speed c * . Turanova obtained uniform L ∞ bounds for the Cauchy problem and deduced the first spreading properties, again for bounded θ [Tur15]. The propagation result was later refined by Bouin, Henderson and Ryzhik [BHR17a] using Turanova's L ∞ estimate. We highlight this point since no uniform L ∞ bound is known for the unbounded case (1.2). In addition, their strategy depended on a "local-in-time Harnack inequality" that is not applicable in our setting. It is also interesting to note that the spreading speed is determined by the linear problem in the case of bounded θ. The same conclusion was drawn by Girardin who investigated a general model which is continuous in the space variable, but discrete in the trait (diffusion) variable [Gir18a,Gir18b].
On the one hand, our work belongs to the wider field of structured reactiondiffusion equations, which are combinations of ecological and evolutionary processes. In a series of works initiated by [ACR13], and followed by [ABR17,BJS16,BM15], various authors studied reaction-diffusion models structured by a phenotypic trait, including a non-local competition similar to (and possibly more general than) −f ρ. However, in that series of studies, the trait is assumed not to impact dispersion, but reproduction, as for e.g. f t = f xx + f θθ + f (a(θ) − ρ). In particular, no acceleration occurs, and the linear determinacy of the speed is always valid. Note that more intricate dependencies are studied, in particular the presence of an environmental cline, which results in a mixed dependency on the growth rate a(θ − Bx), as, for instance, in [ABR17,ACR13] (see also [BM15]).
On the other hand, our work also fits into the analysis of accelerating fronts in reaction-diffusion equations. We refer to [CR12,CR13] for the variant of the Fisher-KPP equation where the spatial diffusion is replaced with a non-local fractional diffusion operator. In this case, the front propagates exponentially fast, see also [Mir20,MM15]. The case where spatial dispersion is described by a convolution operator with a fat-tailed kernel was first analyzed in [Gar11], see also [BGHP18]. The rate of acceleration depends on the asymptotics of the kernel tails. In [BCN15,BCGN16], the authors investigated the acceleration dynamics of a kinetic variant of the Fisher-KPP equation, structured with respect to the velocity variable. The main difference with the current study is that the kinetic model of [BCN15] (see also [CHS12]) enjoys the maximum principle.
A natural question in front propagation, related to the issue of linear determinacy, is whether the long-time and long-range behavior can be described via a geometric equation. This is not always possible -see, for example, [Fre85a, Chapter 6.2] and [MS94,examples B (i) and B (ii)]. In our setting, finding a PDE governing the asymptotic behavior of (1.2) remains an interesting open problem (see the discussion in Section 2).

Notation
We use C to denote a general constant independent of all parameters but θ in and C 0 in (1.3). In addition, when there is a limit being taken, we use A = O(B) and A = o(B) to mean that A CB and A/B → 0 respectively.
We set R ± = {x ∈ R : ±x 0} and R * ± = R ± \ {0}. All function spaces are as usual; for example, L 2 (X) refers to square integrable functions on a set X.
In order to avoid confusion between trajectories and their endpoints, we denote trajectories with bold fonts (x, θ). In general, we use x, x, θ, and θ for points and trajectories in the original variables and y, y, η, and η for points and trajectories in the self-similar variables, which are introduced in Section 4.3.

Approximation of geometric optics from the PDE viewpoint
Our argument follows pioneering works by Freidlin on the approximation of geometric optics for reaction-diffusion equations [Fre85b,Fre86]. In fact, we follow the PDE viewpoint of Evans and Souganidis [ES89]. The approach is based on a longtime, long-range rescaling of the equation that captures the front while "ignoring" the microscopic details. In our case, this involves defining the rescaled functions Note that this scaling comes from the fact that the expected position of the front x(t) scales like O(t 3/2 ) and the expected mean phenotypic trait of the population at the frontθ(t) scales like O(t). We then use the Hopf-Cole transformation; that is, we let Then, after a simple computation, where the complement of the set is taken in R × [h, ∞). Formally passing to the limit as h → 0, we find where the complement of the set is now taken in R × [0, ∞). We note that no bounds on the regularity of ρ are available in the literature so it is not clear that (2.2) can be interpreted rigorously. Another informal candidate for the global limiting Hamilton-Jacobi problem, if any, is the following equation: It is based on the heuristics that the spatial density ρ saturates to the value 1 at the back of the front, i.e. when min θ u(t, x, θ ) = 0, see Figure A.2 for a numerical evidence. However, we are lacking tools to address this issue. Actually, the uniform boundedness of ρ is not yet established to the best of our knowledge (but see [Tur15] for a uniform L ∞ bound in the case of bounded θ). One could also search for an alternative formulation of (2.3) in the framework of obstacle Hamilton-Jacobi equations (see, e.g. [BCGN16,ES89]); however, the non-trivial behavior behind the front complicates this. Nevertheless, in our proof we are still able to use the toolbox of Hamilton-Jacobi equations, such as variational representations and half-relaxed limits (see below) [BP87] in order to reach our goal. We explain in the next paragraph our strategy to replace ρ by a given indicator function µ1 {x<αt 3/2 } . This translates into the equation which admits a comparison principle, together with the variational representation (1.4).

Arguing by contradiction to obtain the spreading rate
Clearly, ρ is the important unknown here. As we mentioned above, no information on ρ other than nonnegativity has been established; even a uniform L ∞ bound is lacking. Thus, it is necessary to take an alternate approach, initiated in [BHR17b]. We argue by contradiction: • On the one hand, suppose that the front is spreading "too fast," that is, at least as fast as α 1 t 3/2 for some α 1 > α * . Roughly, we take this to mean that ρ is uniformly bounded below by 1/2 behind α 1 t 3/2 (the value of the threshold µ = 1/2 matters). With this information at hand, ρ can be replaced by 0 for x > α 1 t 3/2 and by 1/2 for x < α 1 t 3/2 , at the expense of f being a subsolution of the new problem (because the actual ρ is certainly worse than this crude estimate). In other words, we can replace ρ by 1 2 1 {x<α 1 t 3/2 } in the problem (2.2) as in the definition of the variational solution U α, µ (1.4). Therefore, we have lim h→0 u h (1, x, θ) U α 1 , 1 2 (x, θ). Letting h = t −1 1, we thus expect We then notice that by the very definition of α * (1.6). This implies that ρ(t, xt 3/2 ) = ρ h (1, x) is exponentially small around x = α 1 , which is a contradiction. We recall that the derivation of the above Hamilton-Jacobi equation was only formal. To make this proof rigorous, we use the method of half-relaxed limits that is due to Barles and Perthame [BP87]. • On the other hand, suppose that the front is spreading "too slow," that is, no faster than α 2 t 3/2 for some α 2 < α * . Roughly, we take this to mean that ρ is uniformly bounded above by some small δ ahead of α 2 t 3/2 . With this information at hand, ρ can be replaced by δ for x > α 2 t 3/2 and +∞ for x < α 2 t 3/2 , at the expense of f being a supersolution of the new problem (because the actual ρ is certainly better than this crude estimate). In other words, we can replace −1 + ρ(t, x) by −1 + δ + ∞1 {x < α 2 t 3/2 } in the variational problem above. This property implies that lim h → 0 u h (1, x, θ) U α 2 (x, θ) + δ.
Choosing δ small enough, with a similar reasoning as in the previous item, we get another contradiction, since then the front should have emerged ahead of α 2 t 3/2 because min θ U α 2 (α 2 , θ) + δ < 0. While the arguments to prove the emergence of the front are still based on the variational problem, we do not study directly the function u h . Instead, we build subsolutions on moving, widening ellipses -these are actually balls following geodesics in the Riemannian metric associated to the diffusion operator -for the parabolic problem (1.2) following the optimal trajectories in (1.4), using the "timedependent" principle eigenvalue problem of [BHR17b]. Three important comments are to be made. First, this argument is made of two distinct pieces: the rigorous connection between the variational formulation (1.4) and the parabolic problem (1.2) and the precise characterization of U α (α, θ) in order to determine α * . Second, the effective value of ρ is always small on the right side of αt 3/2 in both cases (either 0 or small δ), but it takes very different values on the left hand side (either 1/2 or +∞). Note that ρ is assigned a +∞ value in the absence of any L ∞ bound. At first glance, it is striking that the same threshold α * could arise in both arguments. What saves the day is that the value of U α, µ (α, θ) does not depend on µ provided that µ 1/2. With our method, the latter bound could be lowered at the expense of more complex computations, but certainly not down to any arbitrary small number. Finally, we make a technical but useful comment. To study the variational problem (1.4) we often use the following self-similar variables t = e s , x = t 3/2 y, and θ = tη and study the problem written in terms of such variables. One immediate advantage, beyond the compatibility with the problem, is that µ1 {x < α t 3/2 } , the indicator function in (1.4), becomes µ1 {y < α} , which is stationary. Now, the problem can be seen as the propagation of curved rays in a discontinuous medium with index µ on the left side, {y < α}, and 0 (or small δ) on the right side {y α}.

Optimal trajectories
In the sequel, the qualitative and quantitative descriptions of the trajectories in (1.4) play an important role. We say that a trajectory (x, θ) ∈ A(x, θ) is optimal if it is a minimizer in (1.4) with endpoint (x, θ). We note that the existence and uniqueness of these minimizers is not obvious since the Lagrangian is discontinuous; however, we establish this fact below using lower semi-continuity (see Lemma 3.1).
Why is it that U α, µ (α, θ) does not depend on (large) µ? The answer lies in the optimal trajectories associated with the variational problem (1.4). It happens that the optimal trajectories in (1.4) cannot cross, from right to left, the interface {y = α} if the jump discontinuity is too large (µ 1/2). In short, we prove that the trajectories having their endpoint on the right side of the interface (including the interface itself) resemble exactly Figure 2.1. The nice feature is that they never fall into the left-side of the interface, but they "stick to it" for a while. During the proof, we can, thus, replace the minimization problem (1.4) by the state constraint problem where the curves are forced to stay on the right-side of the interface, so that the actual value of µ does not matter.
In fact, we obtain analytical expressions for the optimal trajectories that lead to the formula for α * involving polynomials of Airy functions. Figure 2.1. Typical optimal trajectories of (1.4) depicted in the self-similar variables plane (y, η) = (x/t 3/2 , θ/t). The endpoint at time t = 1 is (x, θ). The line {y = α} acts as a barrier due to the jump discontinuity in ρ in our argumentation by contradiction. The trajectories with endpoints on the left side of the line {y = α} may come from the right side (not shown). However, the trajectories with endpoints on the right side never visit the left side. Moreover, for times t → 0, they stick to the line {y = α}, together with η → +∞. This behavior holds true if α 4 3 and µ 1 2 .

Evidence for the hindering phenomenon
We now explain why the non-local and local saturation act differently. It is useful to begin by discussing why in the local saturation problem the speed of propagation is determined by the linear problem. Recall that the linear problem is subject to the same asymptotics as (1.4) but with the choice of µ = 0 everywhere, simply because saturation has been ignored. The optimal curves of the linear problem were computed in [BCM + 12]. Rather than giving formulas, we draw them in self-similar variables, see Figure 2.2. Beneath the trajectories, we also draw the zero level line of the value function U 0 , which separates small from large values off (the solution of the linear problem). An important observation is that an optimal trajectory with ending point at the zone wheref is small, remains on the good side of the curved interface at all intermediate times. This means that the trajectory stays in the unsaturated zone where the growth is f (1 − f ) ≈ f . Hence, the trajectory only "sees" the linear problem implying that the optimal trajectories of the linear and the nonlinear problem coincide.
In the non-local problem (1.2), the characterization of the interface does not involve θ. Indeed, the saturated region is given by ρ ≈ 1 and the unsaturated region by ρ 1 (or, better, by min θ U 0 (x, θ) = 0 and min θ U 0 (x, θ) > 0 respectively). However, the trajectories of the linear problem with ending points on the saturated zone cannot remain on the right side of any stationary interface as illustrated in Figure 2.2. Therefore, the saturation term does matter, and it is expected that the location of the interface is a delicate balance between growth, dispersion and saturation. 2), the saturation zone {f δ}, for a small δ > 0, is genuinely a curved area in the phase plane (x/t 3/2 , θ/t). The optimal trajectories associated with (1.4) without saturation (µ = 0) are curved in a similar way. It can be shown that they do not intersect the saturation zone if their endpoint is outside the saturation zone [BHR17b, BMR15]. (B) In the case of a non-local saturation, the saturation zone {ρ δ} is a strip along the vertical direction. The main observation is that the optimal trajectories without saturation (µ = 0) intersect the saturation zone. This yields a contradiction as they are computed by ignoring the effect of saturation. (C) The optimal trajectories of the nonlocal problem with high enough saturation (µ 1/2) do not intersect the saturation zone. Instead, they stick to the interface for some interval of time. The discrepancy between the "local" trajectories (A) and the "non-local" trajectories (C) induces a change in the value function U α , which itself is responsible for the lowering of the critical value from 4/3 (in the local version) to α * ≈ 1.315 (in the non-local version).
This impeding phenomena could be rephrased in a more sophisticated formulation, saying that Freidlin's (N) condition [Fre85b] is not satisfied. Indeed the optimal trajectories of the linear problem ending ahead of the front do not stay ahead at all intermediate times. Therefore, they must have experienced saturation at some time, and so it is not possible to ignore it.
What is more subtle in our case (and leads to explicit results), is that the optimal trajectories of the non-local problem hardly experience saturation, as can be viewed on Figure 2.2 (C): they get deformed by the presence of the putative saturated area, but they do not pass through it so that a uniform lower bound ρ 1/2 in the saturation zone is sufficient to compute all important features explicitly.

Connection with sub-Riemannian geometry
The connection between f and U α that is seen, for example, in (2.4) solicits some comment about a connection with geometry that was first leveraged in [BCHK18]. We ignore the zeroth order term in (1.2) and focus on the diffusion part of the equation. Anticipating the details of the proof in Section 3, let t ∈ [0, T ] for T 1 (fixed), and consider the rescaling t = T 2 τ , x = T 3/2 X and θ = T Θ. Notice the anomalous T 2 in the change of time, so that τ is small, τ 1/T . The diffusion part of the equation (1.2) does not change due to the homogeneity of the second order operator, and the initial data shrinks to the indicator function 1 (−∞, O(1/T 3/2 )] × (0, O(1/T )) . In particular, the problem is not uniformly elliptic in the limit T → +∞. However, it is hypoelliptic in the sense of Hörmander. Moreover, it is a Grushin operator as the sum of the squares of √ Θ∂ X and ∂ Θ respectively. In particular, it satisfies the strong Hörmander condition of hypoellipticity. Therefore, after appropriate rescaling, our problem relies on short time asymptotics of the hypoelliptic heat kernel (2.5). Precise results are known since the 1980's. In particular, from Léandre [Léa87a,Léa87b], see also [BA88], we find where P is the heat kernel associated to (2.5) and dist is the geodesic distance associated with the appropriate sub-Riemannian metric, which coincides with (1.4) up to the zeroth order terms. In particular, we find, at τ = 1/T , which is equivalent to t = T , Notice that this is the formulation of (2.4) but in the absence of reaction terms. In this subsection, we collect some results about U α along with the associated optimal trajectories. In particular, we state two lemmas, which are the main elements of the proof of Proposition 1.1. We also state a proposition that is crucial in the proof of Theorem 1.2. The proofs of these facts may be found in Section 4 and Section 5.
First, we note that minimizing trajectories exist. The uniqueness of the minimizer associated with an endpoint (x, θ) ∈ [α, ∞) × R * + is addressed in Section 6.
Second, we provide a lemma that implies Proposition 1.1.
Next, we show that the optimal trajectories with endpoints on the right side of the front always stay to the right of the front. This is crucial, since, if this were not true, a uniform upper bound on ρ would be required in order to proceed.
For the purposes of the proof in the next section, we also mention a technical result that is established after a careful description of the minimizing trajectory associated with any endpoint (α, θ). We show (cf. Lemma 7.10) that the optimal trajectories are such that, for t 1, We make two comments. First, such an anomalous scaling is not obvious at first glance. In fact, it arises when the optimal trajectory comes into contact with the barrier {x = αt 3/2 }. Second, we do not believe such an elaborate result is required in the proof in the following section; however, as the result was readily available, we use it.

Proof of the lower bound in Theorem 1.2
The proof of the lower bound (1.7) in Theorem 1.2 follows almost along the lines of the work in [BHR17b].

Proof.
Step 1. -Definition of some useful trajectories. Fix > 0. Using the definition of α * and Lemma 3.2, there exists r > 0 and x, θ, depending only on , such that One may worry about the behavior of the integral as s 1, but we see that the peculiar behavior (3.1) guarantees that L α * − 2 (s, x(s), θ(s),ẋ(s),θ(s)) is integrable at s = 0. By a density argument, up to reducing the value of r > 0, we may assume that x, θ ∈ C 2 ([0, 1]), keeping the behavior θ(t) Ct for some constant C > 0 as t → 0 (the asymptotics for t 1 for the regularized trajectory are possible due to (3.1)). In addition, from Proposition 3.3 we get that For T > 0, x 0 ∈ R, θ 0 1, and t ∈ [0, T ], define the scaled functions, The parameters x 0 and θ 0 are determined in the sequel. For notational ease, we refer to X T, x 0 and Θ T, θ 0 simply as X and Θ in the sequel. Noticing that Θ T θ( t T ), then changing variables in (3.2) from the definition of L α * − 2 (1.5), we get the crucial fact, Further, we may assume without loss of generality which only decreases the left hand side of (3.5). In fact, the above argument shows that the optimal trajectory associated with the minimum value of U α (with respect to θ) is decreasing in its second argument. We note, however, that it is not true that all optimal trajectories are non-decreasing in the second argument.
Step 2. -A subsolution in a Dirichlet ball along the above trajectories. Let δ = r/3. We now argue by contradiction. Assume that (1.7) does not hold. Then there exists t 0 such that, for all t t 0 , We may assume, by simply shifting in time, that t 0 = 0 and that f 0 is positive everywhere. Further, using (3.6), we have, Next we find a subsolution of (3.7) in a Dirichlet ball that moves along the above trajectories. To this end, we define, for any (x, θ) and R, We use the following lemma, which is very similar to [BHR17b, Lemma 4.1] (see also [BCHK18,Lemma 13]). Its proof is postponed, but we use it now to conclude the proof of the Theorem 1.2.
Next, we find x 0 that is independent of T such that, for all t ∈ [0, T ], Then, Hence, for (3.9) to hold it is enough to show that, for all t ∈ [0, T ], From (3.3) and (3.4), we see that Since θ is Lipschitz continuous, then there exists a constant A, independent of t and T , such that Θ(t) At + θ 0 . Thus, we can choose x 0 large enough, independently of t and T , such that The combination of this and (3.11) implies (3.10), and hence (3.9) holds.
Step 3. -Obtaining a contradiction. With the choice of x 0 such that (3.9) holds, we then define and the subsolution v β = βv given by Lemma 3.4. According to (3.9) and (3.7), f is a supersolution to the linear parabolic equation satisfied by v β in (3.8). In addition, The previous line, together with the definitions of ρ and E X(t), Θ(t), R/2 , yields, As the constant ω(R) depends only on R, we can enlarge the value of T such that ρ(T, X(T )) βω(R)Re δT 2δ. This is a contradiction, as the combination of (3.6) and (3.9), evaluated at t = T , implies that ρ(T, X(T )) < δ. Finally, we establish Lemma 3.4. The proof is very similar to those of [BHR17b, Lemma 4.1] and [BCHK18, Lemma 13]; however, it does not immediately follow from either, so we provide a sketch. To this end, we need the following auxiliary lemma. , There exists a constant C (δ) such that, for all R C (δ), θ 0 C (δ), and T C (δ), then there is a function w(t, y, η) satisfying where ω (R) depends only on R.
Proof. -This is essentially a restatement of [BHR17b, p. 745], which in turn uses [BHR17b, Lemmas 5.1, 5.2]. What we denote w here is denoted by w T, H in [BHR17b]. The only thing we need to verify is that the hypothesis of [BHR17b, Lemma 5.1] holds in our situation. That is, we must verify We show that the second term in A converges to zero; the rest are handled similarly (in fact, more easily). Using the definitions of X and Θ (3.4), we find, Next, according to the choice of the reference trajectory (x, θ), there exists a constant C such that When t/T is small, we use Young's inequality to see that t 1/3 θ Notice that this tends to zero as θ 0 → ∞. When t/T is away from 0, θ(t/T ) is uniformly strictly positive, and so (3.16) converges to zero as T → ∞. Proof of Lemma 3.4. -Before beginning, we point out that, using (3.17), as in the proof of Lemma 3.5, we find that there exists a constantC(R) that depends only on R such that Let w be as given by Lemma 3.5. Define, for (y, η) A direct computation, together with the fact that w is a subsolution of (3.12), shows In addition, according to (3.13) we have v ≡ 0 on ∂B R (0, 0). Also, by the definition of v, the fact that g(0) = 0, (3.14), and (3.18), we have, for (y, Next we find a lower bound for v(T, y, η) on B R/2 (0, 0), for which we first bound g(T ) from above. Using (3.5), it follows that Applying again (3.17) in the manner of the proof of Lemma 3.5, there is another constantC(R) (that we do not relabel) such that, Together with (3.18), the definition of v, and (3.15), we find, for (y, Finally, we recover that v is the desired subsolution of (3.8) by making the change of variables from v to v in the moving frame; that is, we let We recover the last conclusion in Lemma 3.4 by letting ω(R) = ω (R)e −C(R) , concluding the proof.

Proof of the upper bound in Theorem 1.2
Proof. -We wish to prove by contradiction that, for all > 0, Suppose on the contrary that there exists > 0 and t 0 such that, for all t t 0 and all x (α * + )t 3/2 , In this case, we see that, for all t t 0 , The work in [BHR17b, Section 3] implies that there exists a constant C, depending only on C 0 in (1.3), such that Here ψ is a positive function, defined piecewise in [BHR17b, Section 3], whose exact form is unimportant, but which is positive when max{x, θ} > 0 and satisfies, for any We use this particular scaling for two purposes. First, up to shifting in time, we may assume, without loss of generality, that t 0 = 0 and that there is a constant C 0 > 0 such that, for all (x, θ) ∈ R × (1, ∞), Second, for any small parameter h > 0, we define Then, due to (3.21), u h satisfies both the bound and, due to (3.20), the equation We define the half-relaxed limit u * = lim sup h → 0 u h . We claim that, for any δ > 0, u * satisfies (in the viscosity sense) We point out that we have reduced to /2. The first two inequalities follow from standard arguments in the theory of viscosity solutions, see, e.g., [HPS18, Section 3.2] for a similar setting. The third inequality follows directly from the upper bound (3.23) and the fact that ψ(x, θ) is positive for max{x, θ} > 0. The restriction to the outside of a ball of radius δ (for arbitrary δ > 0) might look unnecessary. However, in [CLS89], which is applied in the sequel, only "maximal functions" with support on smooth, open sets are considered. Using (3.24) along with theory of maximal functions [CLS89] (see also [HPS18] for a discussion of the boundary conditions and the degeneracy in the Hamiltonian near the boundary {θ = 0}, both of which are not considered in [CLS89]), along with the Lax-Oleinik formula [Lio82, Chapter 11], we see, for all (x, θ) ∈ R × R + , Taking the limit δ → 0 and setting t = 1, we find Fix any x α * + /2. Using Proposition 3.3, the trajectory (x, θ) satisfies x(s) For notational ease, let r = min θ U α * + /2 (α * + /2, θ ). According to Lemma 3.2 and the definition of α * , we have r > 0; thus, we find We now use the negativity of u * to show that f is small beyond (α * + )t 3/2 for large times, which provides a contradiction. From the definition of u * , it follows that there exists h 0 > 0 such that if h h 0 , then, for all x ∈ (α * + 2 /3, 2) and all θ ∈ (h, 4), Hence, if t 1/h 0 , x ∈ ((α * + 2 /3)t 3/2 , 2t 3/2 ) and θ ∈ (1, 4t), then which implies, On the other hand, by [BHR17b, Equation (3.5)], there exists a positive constant C, depending only on the initial data f 0 such that f (t, x, θ) Ce t−θ 2 /4t , for all (t, x, θ). Hence, To obtain the final inequality, we have used that ∞ z e −θ 2 /z dθ e −z /2 holds for any z > 0. This fact can be verified as follows, using first the fact that 1/θ is decreasing and then a change of variables: The combination of (3.25) and (3.26) implies To rule out the other part of the domain, we apply [BHR17b, Theorem 1.2], which implies, Combining these two estimates yields This contradicts (3.19), since the latter condition implies that The proof is complete.

Basic properties of the minimizing problem U α, µ
In this section we prove some basic properties of the trajectories. Namely, we give the proofs of Lemma 3.1, and Lemma 3.2. We also conclude with the reformulation of the minimization problem in the self-similar variables.

The existence of a minimizing trajectory -Lemma 3.1
The existence of minimizers is a delicate issue due to the discontinuity in the Lagrangian L α, µ . From our qualitative analysis in the sequel, we show that optimal trajectories eventually stick to the line of discontinuity for periods of time. Therefore, the value of the Lagrangian on this line matters. As an illustration of the subtlety of this issue, notice that replacing 1 {x < αt 3/2 } by 1 {x αt 3/2 } would break down the existence of minimizers. In the latter case, a minimizing sequence would approach the line without sticking to it (details not shown).
Proof. -Take any minimizing sequence (x n , θ n ) ∈ A(x, θ) such that We now establish that x n and θ n are bounded in H 1 ((0, 1); R × R + ), uniformly in n.
Without loss of generality, we take n large enough so that, We use the previous line and the definition of L α, µ to find, For t ∈ [0, 1], we have, where the last inequality follows from (4.1). Using the previous line to bound the first term in the integrand on the left-hand side of (4.1) yields, Thus x n and θ n are bounded in H 1 ((0, 1); R × R + ), uniformly in n. Due to the uniform H 1 bound on (x n , θ n ), we find, up to extraction of a subsequence, (x n , θ n ) (x, θ) for some trajectory (x, θ) ∈ H 1 . This convergence is strong in C 0 due to the Sobolev embedding theorem. In order to take advantage of this extraction, we examine the properties of L α, µ . We split it into two parts: is convex (possibly taking value +∞), as can be seen readily from the Hessian of the function F (v, θ) = v 2 /(4θ), namely combines a contribution that involves derivatives, which is convex, and a contribution that involves x only, which is genuinely lower semi-continuous.
We now establish the lower semi-continuity of both terms, beginning with the first term. We claim that this term is lower semi-continuous with respect to the strong topology; this is postponed until the conclusion of this proof. Then, due to [Bre11, Corollary 3.9], which states that convexity and lower semi-continuity in the strong topology (for H 1 ) implies lower semi-continuity in the weak topology, we conclude that We now consider the second term. This contribution is lower-continuous for the topology of strong convergence by Fatou's lemma, hence lim inf We conclude by merging both contributions, that is The last inequality follows from the definition of U α,µ . Hence, the inequalities must all be equalities above, implying that (x, θ) is truly a minimizing trajectory.
The proof is now concluded after showing the lower semi-continuity of the first term with respect to the strong topology. We fix terms carefully here. Define We show the lower semi-continuity of I. Fix any x n , θ n , x, θ ∈ H 1 such that x n → x and θ n → θ in H 1 . Note that this implies that θ n → θ in C 0 , which we use strongly below. Fix any > 0. Then Sinceẋ n →ẋ andθ n →θ in L 2 and since θ −1 Applying the monotone convergence theorem, we find Hence, I is lower semi-continuous as claimed.
Define s h = sup{s : x h (s) = x}. Notice that s h is well-defined due to the continuity of x h established above, along with the fact that We construct a trajectory connecting the origin and the point ( Using these two facts to bound the right-hand side of the last line in (4.2) from above yields finishing the proof that U α is strictly increasing with respect to x α. We now prove that min θ U 4/3 (4/3, θ) > 0. For this, we first recall the particular trajectories that were computed in [BHR17b], in the case without growth saturation, i.e., when α = 0 (those computations were originally derived for [BCM + 12], though they are not explicitly written there, so we provide [BHR17b] as a reference instead). It was shown that the minimum of U 0 (4/3, ·) is reached at θ = 1, with U 0 (4/3, 1) = 0. Let (x 0 , θ 0 ) be the optimal trajectory associated with the endpoint (4/3, 1). Then, x 0 has the following simple expression: A crucial observation is that x 0 is always to the left of the barrier associated with α = 4/3, i.e., (4.3) x 0 (t) < 4 3 t 3/2 for all t ∈ (0, 1). Indeed, Next, let (x, θ) be a minimizing trajectory associated with α = 4/3, that is,

Reformulation of the minimization problem in the self-similar variables
In (3.4) and (3.22) we use the scaling properties of our problem. Here, we go a step further, as we reformulate the minimization problem (1.4) in self-similar coordinates. We transform each trajectory (x(t), θ(t)) for t ∈ (0, 1) into the new (y(s), η(s)), s ∈ (−∞, 0) as follows Note that the endpoint is not changed: (y(0), η(0)) = (x, θ). The minimization problem (1.4) is equivalent to the following one: (4.5) η(s),ẏ(s),η(s)) e s ds : (y(·), η(·)) ∈ A (x, θ) , where the autonomous Lagrangian L α, µ is given by and the set of admissible trajectories is given by In view of the discontinuity in the Lagrangian along the line {y = α}, we expect interesting dynamics as y(s) approaches α. We prove in the next section that the line {y = α} acts as a barrier for the optimal trajectories that end in the area {y α}, provided that µ is not too small and α is not too large, as stated in Proposition 3.3.
Due to the natural scaling of the problem, it is often convenient notationally to let (4.8) α = 3α 4 .

Qualitative properties of trajectories -Proposition 3.3
The next result is a reformulation of Proposition 3.3 using the self-similar coordinates introduced in Section 4.3.
Lemma 5.1. -Suppose that 2µ α 4/3 . Let (x, θ) ∈ R × R * + be an endpoint such that x α. Then any optimal trajectory (y, η) ∈ A (x, θ) of (4.5) satisfies y(s) α for all s ∈ (−∞, 0]. That is, if y ends beyond the line {y = α}, then it never crosses the line. It is clear that this is a consequence of the following two lemmas.    If (y, η) is an optimal trajectory for (4.5), then η is nonincreasing over (−∞, 0). The proof of Lemma 5.4 is a direct consequence of the Hamiltonian dynamics associated with (4.5). We review it briefly in the next section. The other two statements require additional conditions on α and µ, as in Lemma 5.1. They are proved in Section 5.2.

A brief overview of Hamiltonian dynamics
In this section we provide some elements of the computation of the optimal trajectories that we use in the article. To this end, it is instructive to briefly recall the basics of calculus of variations in a smooth setting. Let L(X, V ) be some smooth Lagrangian function. Consider, for some admissible set of trajectories A with endpoint at x ∈ R d , the following problem: We begin with the following important property of optimal curves. This will be used repeatedly to restrict the arguments to portions of curves in the sequel.
Lemma 5.5 (Optimality on sub-intervals). -Suppose X is an optimal curve for the minimizing problem (5.1), and let s 0 0. Then, the restriction X(s + s 0 ) for s < 0 is also an optimal curve for the problem with endpoint at X(s 0 ).
Proof. -Suppose that X(· + s 0 ) is not optimal. There exists Z with a strictly smaller cost such that Z(0) = X(s 0 ): or, equivalently, after translating time and using specifically the exponential weight, From our choice of P along with the representation formula for L, we see that H(X, P ) + L(X, D P H(X, P )) = P · D P H(X, P ), so that the above becomesḢ + H + L = 0, or, equivalently, We deduce from (5.3) and (5.1) that U (X(0)) = −H(X(0), P (0)). Finally, we point out a nice relationship between D X U and P : Indeed, if we perturb the optimal trajectory X by a constant velocity V on the last portion of the time interval (− , 0), we find by the minimization property (5.1): Next, dividing both sides by 2 , then letting → 0, we find that which is equivalent to (5.4) by definition. In our setting, the Hamiltonian associated with (1.4), is This follows from (4.6), where we solve for the Lagrangian. Thus, the Hamiltonian system (5.2) is, for the portion of the trajectories on either of the half-spaces {y < α} and {y > α}, Here we use the fact that 1 {y < α} is constant on each half space. The connection between the two half-spaces must be handled with care, see below for details. The general solution of (5.6) on any interval of free motion, i.e. avoiding the line {y = α}, for trajectories ending at (x, θ) at s = 0, is, for some constants A and B, Due to (4.6) and (5.6), the running cost on each half-space {y < α} and {y > α} is then given by: L α, µ (y(s), η(s),ẏ(s),η(s)) = η(s)|p(s)| 2 + |q(s)| 2 − 1 + µ1 (−∞, α) (y(s)) .

ANNALES HENRI LEBESGUE
An immediate computation yields that this quantity is constant on each half-space {y < α} and {y > α}. In particular, on some interval (s 0 , 0) such that y(s) stays on the same side of the line, the running cost is η(s),ẏ(s),η(s) We now investigate the portions of (y(s), η(s)) when y(s) = α for an open interval of time s ∈ (s 1 , s 0 ). It is convenient to extract the dynamics from the Lagrangian function (4.6) when the trajectory has been confined to the line. When confined to this line, the Lagrangian is which is obtained from (4.6) by setting v y = 0, and y = α, and µ1 (−∞, α) (y) = 0.

Better stay on the right side -Lemma 5.1
We now establish that any trajectory that ends to the right of the line {y = α} must always be to the right of this line. Our approach, in each lemma, is a careful analysis of the minimizing trajectories, which we can write down semi-explicitly thanks to the computations performed in Section 5.1. In each case, we show that, were the undesired behavior to occur, we may construct a related trajectory with a lower cost, contradicting the fact that the offending trajectory was a minimizer.
We first prove the monotonicity of optimal trajectories in η. This is an important step in establishing Lemma 5.3.
Proof of Lemma 5.4. -Let (y, η) ∈ A (x, θ) be the optimal trajectory. Note that, by definition of y, η in terms of x, θ, and the fact that x, θ ∈ H 1 ((0, 1); R × R + ) (see proof of Lemma 3.1), we deduce y, η ∈ H 1 loc ((−∞, 0); R × R * + ). We begin by obtaining a differential inequality for the second derivativeη in the distributional sense. We note that we have not established the continuity ofη or any regularity of η, so we are forced to work with this distributional inequality.
Fix any > 0 and any 0 φ ∈ C ∞ c (R * − ). Notice that (y, η + e −s φ) ∈ A (x, θ). Thus, we have, µ (y, η,ẏ,η) Writing out the expressions and re-arranging the terms, we see, Expanding the first term and dividing by yields, Applying the monotone convergence theorem, we get, Since this is true for all φ, it follows thatη +η 0 in the sense of distributions, from which it follows that d ds (e sη ) 0 holds in the sense of distributions. We now conclude the proof by choosing an appropriate test function. If η is not non-increasing, then there exists a 0 ψ ∈ C ∞ c (R * − ) such that ηψe s ds = γ > 0 and ψds = 1. Fix any s < inf supp(ψ) and > 0 such that < −s . Let φ be a standard mollifier with supp φ ⊂ (− , + ). Then, define the smooth test function Note that from our choice of s and , the above test function is positive and compactly supported in R * − . From our choice of φ and ψ along with the differential inequality established above, α for all s ∈ (s 0 , 0), and let denote θ 0 = η(s 0 ). By Lemma 5.5, we can assume without loss of generality that (y(0), η(0)) = (α, θ 0 ) and that (y, η) is a minimizing trajectory in A (α, θ 0 ). By assumption, we note that y(s) < α for all s < 0. Then, it is a global solution of the system (5.6) with x = α.
From (5.7) we have for some A, B ∈ R. On the one hand, multiplying the first and second equality in the previous line by, respectively, e s and e 3s 2 , and then taking the limit s → −∞ and using the conditions in (4.7) implies the following equations: which is equivalent to Since A → θ 0 A + 1 3 A 3 is increasing, A is uniquely determined. On the other hand, computingẏ(0) and using the conditionẏ(0) 0 implies where we recall that α = 3α/4. Finally, from (5.8), the global cost of the trajectory equals the (constant) running cost: This global cost can be compared with the cost of the steady trajectory located at the same endpoint. Indeed, let ( y(s), η(s)) = (α, θ 0 ) for all s ∈ (−∞, 0). It is clear that ( y, η) ∈ A (α, θ 0 ). From (5.9), the associated cost is This trajectory is by no means globally optimal; however, it has a lower cost than the trajectory (y(s), η(s)) under the assumptions of Lemma 5.1. Indeed, we wish to show that L α, µ (y, η,ẏ,η) > L α, µ ( y, η,ẏ,η), which contradicts the fact that (y, η) is a minimizing trajectory. This is equivalent to showing that (5.14) According to (5.12) and (5.13), we have the following constraints on the values of A: This suggests that we use the new variables a and b such that According to the definition of a, and by (5.13) and (5.15), we have a ∈ 1, 4 3 and b ∈ [0, 1] .
With the definitions of a and b, the inequality (5.14) is equivalent to We now prove (5.16), which finishes the proof. Since b 3 = 4 − 3a, then The right hand side of this expression is increasing for a ∈ (1, 4 3 ): its derivative with respect to a is Thus, we may bound the right hand side of (5.17) by its value at a = 1, which implies that Hence, we obtain α 4/3 µ where we used the condition α 4/3 < 4µ in the last inequality. Hence, we have established (5.14), contradicting the fact that (y, η) is a minimizing trajectory. This concludes the proof of Lemma 5.2. Note that we have used the weaker condition α 4/3 < 4µ instead of α 4/3 2µ. In fact the next proof requires a more stringent condition on the parameters.
Proof of Lemma 5.3. -To proceed with the non-optimality of the C-turn, we make the following reduction. As above, by Lemma 5.5 we may suppose, without loss of generality, that s 0 = 0 and s 1 < 0 (see Figure 5.1).
Since the trajectory does not cross the line {y = α} during the time interval (s 1 , 0), the optimal trajectory (y, η) is given by (5.7), with x = α, for some constants A, B ∈ R. We point out that, by Lemma 5.4, Further, since y(0) = α and y(s) < α for s ∈ (s 1 , 0), it follows thatẏ(0) 0. Hence (5.13) is valid. There seems to be no natural way to compare the trajectory with a steady trajectory as in the proof of Lemma 5.2. Alternatively, we compare the trajectory (y(s), η(s)) to the trajectory ( y(s), η(s)), where we define In short, ( y, η) is obtained by projecting the portion between s 1 and 0, the C-turn, onto the line. It is clear that ( y, η) ∈ A (α, θ 0 ). To show that ( y, η) has a lower cost than (y, η), it is enough to compare the partial costs on the interval (s 1 , 0). The cost for (y, η) is, via (5.8), The cost of ( y, η) on (s 1 , 0) is, where we have obtained the second equality by using the expression for η in (5.7) and computingη. We now consider the difference J orig − J new . The above formulas imply where to obtain the last equality we have used the expression for η in (5.7). Since the integrand is increasing with respect to η, it is fruitful to bound η(s) from below. In view of (5.7), this amounts to bounding B from above. In parallel with the proof of Lemma 5.2, we shall use the information at s = s 1 in order to gain an estimate for B. Evaluating at s 1 the expression for η in (5.7), and then using (5.18), yields Notice (2 − e s 1 − e −s 1 ) = (e −s 1 − 1)(e s 1 − 1). Using this, along with the bound above, we see, for all s ∈ (s 1 , 0), (5.20) Let In view of (5.20), along with (5.19), we find where we have used the bound (5.20) for the first occurrence of η(s) in (5.19), but the less precise estimate η(s) θ 0 for the second occurrence. Thus, in order to control the sign of J orig − J new , it is sufficient to show I > −µ.
We now establish the lower bound on I. An explicit computation yields Recall that, due to (5.13), θ 0 A α. Hence, The quantity on the right hand side is minimized (with respect to θ 0 ) when θ 3 0 = 4α 2 (1 − e s 1 )/3. Thus, we have Recall that α 4/3 2µ. Also, notice that 1 − e s 1 1. Hence, In view of the definition of I, this implies that J orig − J new > 0. Thus (y, η) cannot be a minimizer, since the cost of ( y, η) is strictly smaller. This concludes the proof of Lemma 5.3.

The explicit characterization of α * -Theorem 1.3
En route to proving Theorem 1.3, the exact shape of the function U α must be deciphered, at least when restricted to endpoints (α, θ). This involves a careful handling of the connection between the portions of the trajectory that move freely in (α, ∞) × R * + , and those that stick to the line {y = α}. In order to begin the discussion, we first establish the uniqueness of optimal trajectories. The proof also establishes the convexity of U α (x, θ) on the domain [α, ∞) × R * + . This is the content of the following lemma.
Knowing that optimal trajectories are unique allows us to completely characterize them. This characterization relies on good properties of an auxiliary function Q : R * + → R, (see Section 7 for a precise definition). Here, we rely only on the useful properties that Q(θ) θ is strictly increasing, and therefore (6.1) there exists a unique θ such that Q(θ ) = θ 4 , which separates those trajectories that make an excursion to the right versus those that "stick" to the line {y = α} (see Proposition 6.2 (iii) for a precise statement of the latter property). From Q and θ , we also define the entire family: where recall from (4.8) thatᾱ = 3α/4. In particular, we have Q ≡ Q 4/3 . The next Proposition 6.2 gathers useful properties of the optimal trajectories. Useful notations are illustrated in Figure 6.1 for the reader's sake.
Proposition 6.2. -xLet θ > 0 and let (y, η) be the optimal trajectory in A (α, θ). Then (y, η) satisfies the following conditions: We postpone the proof of this important list of results to Section 7. However, we can make a few comments about some quantitative statements there. Firstly, we find that the optimal trajectory sticks precisely to the line {y = α} for some interval (−∞, s ], with s 0 (in fact, s = 0 if and only if θ θ ). We refer to s as the contact time. Secondly, and quite importantly, q and η are linked by the relationship Proposition 6.2.(v) when s s . It turns out that the constraint at s = −∞, η(s) = o(e −s ), selects one branch of the family of solutions, and we can identify and describe this explicitly using the identity q = Q α (η).
To be able to identify the contact time with an analytical equation, we also need the following technical lemma on real functions, which is going to be used with the change of unknown τ = exp(s/2) = t 1/2 ∈ (0, 1). Lemma 6.3. -Let R be defined by (1.8), and V be defined by (1.9). Define the function ξ by if and only if hold. Moreover, there is at most one τ 0 ∈ (0, 1) such that (6.4) holds. The uniqueness of τ 0 is proved by a monotonicity argument: dividing each side of the equation by ξ(τ ), we find that the left-hand side V (τ ) 2 /ξ(τ ) and the righthand side Q(ξ(τ ))/ξ(τ ) have opposite monotonicity, as illustrated in Figure 6.2. The difficulty arises in showing the monotonicity of θ → Q(θ)/θ.
We prove Lemma 6.1, Proposition 6.2 and Lemma 6.3 in Section 7. We now show how to conclude Theorem 1.3 from these three results.

The homogeneity of U α -Theorem 1.3 (i)
We begin by establishing the homogeneity of U α . While this neither relies on Lemma 6.1, Proposition 6.2, nor Lemma 6.3, it is used to establish Theorem 1.3 (ii).

The analytical value of α * -Theorem 1.3 (ii)
In this subsection, we show how to get an algebraic equation for α * . In order to compute this value, due to Theorem 1.3 (i), it is enough to fix α = 4/3, and to compute min θ U 4/3 (4/3, θ). We first show that such a minimum is attained at a unique point θ min . We then identify the optimal trajectory ending at (4/3, θ min ). The identification of this trajectory relies on the computation of the contact time s (θ min ). Once the optimal trajectory is characterized, we can compute the value of U 4/3 (4/3, θ min ), hence U α (α, θ min ) by homogeneity. Figure 6.3 represents the function U α (α, ·) at α = α * , for the sake of illustration. Proof of Theorem 1.3 (ii).
Indeed, if we have a point θ 0 such that (6.7) holds, and (y, η) is the optimal trajectory with endpoint at (4/3, θ 0 ), we find, where the second equality follows by (5.4). This is precisely the characterization of a critical point of U 4/3 (4/3, ·), implying that θ 0 is indeed the unique minimum of U 4/3 (4/3, ·) as desired. We now prove that there is indeed such a point θ 0 satisfying (6.7). Note first that by Proposition 6.2 (iii), θ < θ implies that s (θ) < 0. Moreover, by Proposition 6.2 (vi), we obtain that q(s) = B + A 2 (1 − e s ) holds for s ∈ (s , 0]. Thus, q(0) = 0 is equivalent to B = 0, which, by Proposition 6.2 (vi) is equivalent to, We use the intermediate value theorem to find a θ 0 ∈ (0, θ ) satisfying (6.8). For θ = θ , we have s (θ ) = 0, and η(s (θ )) = η(0) = θ , by Proposition 6.2 (iii). Hence, the left hand side of (6.8) is Q(θ ) = θ /4 > 0 (we have used (6.1), the definition of θ ), whereas the right hand side is zero. Next, we show that the left hand side is smaller than the right hand side as θ → 0. To this end, we use the combination of (5.7) (at s = s ) and Proposition 6.2 (vi) to get, Therefore, we have: To conclude, it is enough to show that the right hand side in the latter expression has a positive limit as θ → 0. To this end, we first claim that lim inf θ → 0 s (θ) < 0 holds. Indeed, if this were not the case, then we would find, using the continuity of η, that lim inf θ→0 η(s (θ)) = 0, which is impossible, according to Proposition 6.2 (ii). Therefore, we find that the right hand side of the previous line indeed has a positive limit as θ → 0, and therefore, in view of the discussion above, an interior minimum occurs at some θ min ∈ (0, θ ) since we can solve (6.8).
Step 2. -# Identification of the contact time s (θ min ): Letting (y, η) be the optimal trajectory ending at (α, θ min ), we have an explicit expression for (y, η) in terms of A and B. In addition, we know B = 0 from the discussion preceding (6.8). For notational ease, let θ = η(s ), and τ = e s /2 ∈ (0, 1). (The latter is the square root of the contact time in the original t variable). We shall show that τ is exactly the τ 0 defined by (1.11).

Characterizing the optimal trajectories
Proposition 6.2 is proved piecemeal throughout the sequel. We do not make note immediately when any portion is proved. Instead, we compile the proof in the last Section 7.6.

Uniqueness of minimizing trajectories -Lemma 6.1
We switch back to the original variables for the proof of this lemma.
The strict convexity of U α follows immediately from a similar argument. As such, we omit it.

Trajectories have at most one interval of free motion
We refer to each portion of the trajectory not intersecting the line {y = α} as "free motion." A preliminary observation is that free motion for all time is not permitted.
Proof. -We proceed by contradiction. Suppose that y(s) > α for all s < 0. By (5.7), we have both, as s → −∞, The growth conditions in the definition (4.7) of A (α, θ) imply, Returning to (7.3), these conditions imply the strong asymptotic behavior y(s) = O(e s/2 ) → 0, as s → −∞. This obviously violates the hypothesis that y(s) > α for all s. We now investigate the dynamics of a trajectory as it comes into contact with the line. If s 0 < 0 is a contact time, we expect thatẏ(s 0 ) = 0, since y(s 0 ) = α is a local minimum. To obtain this, we need to establish sufficient regularity of the optimal trajectories. From (5.6), this allows us to define p(s 0 ) = α/η(s 0 ) in a continuous way. Regularity is the purpose of the next statement. The proof relies on a preliminary Lipschitz bound on η. We state and prove this now, and then continue with the proof of Lemma 7.2. Lemma 7.3 (Lipschitz bounds on η). -Under the assumptions of Lemma 7.1, η ∈ W 1, ∞ loc (−∞, 0). In addition, q is locally bounded. Proof. -We begin by smoothing the Lagrangian, in order to use classical theory. For any ∈ (0, 1), let It is non-negative, convex, twice differentiable, and it takes value 0 if y α. Then define The Lagrangian L α approximates L α, +∞ with the state constraint condition that trajectories must lie on the set {y α}. Let us consider the variational problem associated with L α , with endpoint (α, θ), in the original variables. Following an argument similar to the proof of Lemma 3.1, one can prove that there exists a minimizing trajectory (x , θ ) to this variational problem. Moreover, such minimizing trajectories are bounded in H 1 ((0, 1); R × R + ), uniformly in . It follows that (x , θ ) converges, along subsequences, strongly in L 2 and weakly in H 1 to a trajectory (x, θ), which is the minimizing trajectory associated with L α . We next go back to the variational problem with self-similar variables. Let the trajectories (y , η ) and (y, η) be the trajectories in the self-similar formulation of the problem corresponding respectively to (x , θ ) and (x, θ). Note that thanks to the bounds on (x , θ ) and (x, θ) we deduce that (y , η ) and (y, η) are bounded in H 1 loc ((−∞, 0); R × R + ), uniformly in .
Since L α is smooth, we use (5.6) to write the Hamiltonian system for (y , η , p , q ): Since χ is regular, each of the quantities above is well-defined. Further, we obtain η +η = 2q .
First we show that q ∈ W 1, 1 loc (R * − ), with bounds in this space independent of . In the sequel, by saying that a sequence f is "bounded uniformly in X loc ," we mean that for every compact set K ⊂ (−∞, 0], there is a constant C K , depending only on K, such that f X(K) C K for all ∈ (0, 1). From (7.4), we get From the first contribution in the formula of L α and the fact that (y , η ) is a trajectory minimizing a variational problem similar to (4.5) but replacing L α by L α , it follows that e s/2 √ η p is bounded in L 2 uniformly in . An argument similar to the one in the proof of Lemma 5.4 shows thatη 0 and, hence, η (s) θ for all s. In fact, this is easier to prove since (7.4) is a smooth Hamiltonian system. From the above, we conclude that p is uniformly bounded in L 2 loc . This, in turn, implies thaṫ q is uniformly bounded in L 1 loc , by the last identity in 7.4. From the second contribution in the formula of L α , we deduce that ∂ s (e s η ) = 2e s q (s) is bounded uniformly in L 2 loc , and, thus, in L 1 loc as well. First, we conclude that q belongs to W 1, 1 loc as claimed above. Hence it is locally uniformly bounded, independently of , and so is q after taking the limit → 0. Next, it follows from differentiating (7.4) and using the bound onq that ∂ s (e sη ) is bounded uniformly in L 1 loc . Lastly, we observe that e s η is bounded uniformly in L ∞ loc as a consequence: Combining all three bounds, we see that e s η is uniformly bounded in W 2, 1 loc . The Sobolev embedding theorem then implies that e s η (s) is uniformly bounded in W 1, ∞ loc . From this, it follows that η andη + η are bounded uniformly in L ∞ loc . Taking a linear combination of these two locally bounded functions, we find thatη is uniformly bounded in L ∞ loc . Passing to the limit → 0 yields the local Lipschitz bound on η, and the proof is completed.
With the local L ∞ bound onη from Lemma 7.3, we are ready to tackle of the continuity ofẏ.
Proof of Lemma 7.2. -We follow the same lines as in the previous proof. However, we differentiate the first equation in (7.4) so as to get: All terms on the right hand side are bounded except the last one. To handle it, we multiply byÿ /η on both sides to get where f is uniformly bounded in L 2 loc (R * − ). As noted above, η is non-increasing. Therefore, dividing by η on a compact sub-interval of R * − is not an issue. To conclude, multiply by a given test function φ ∈ C ∞ c (R − ), and integrate by parts: We conclude by noticing that 0 −∞ χ (y )φ ds is bounded uniformly in as it appears in integral form in the variational problem associated with L α . As a result,ÿ is in L 2 loc independent of . Therefore, y is indeed bounded in H 2 loc ((−∞, 0); R × R + ), uniformly in . Passing to the limit, we get that y is also in H 2 loc , which implies thanks to the Sobolev embedding theorem that y ∈ C 1, 1 2 loc (−∞, 0). Now, if s 0 < 0 is such that y(s 0 ) = α, thenẏ(s 0 ) = 0 since s 0 is the location of a minimum of y. On the other hand, since (5.6) is satisfied whenever y(s) > α, then we find, This concludes the proof of the continuity of y andẏ. The continuity of q is a consequence of this. Indeed, the local boundedness of y andẏ , (7.4), Lemma 7.3, and the fact that η is non-increasing imply that p is uniformly bounded in L ∞ loc . Sinceq = −|p | 2 , thenq is uniformly bounded in L ∞ loc as well. This, together with the fact that η is uniformly bounded in W 1, ∞ loc (see the proof of Lemma 7.3), implies that q is uniformly bounded in W 1,∞ loc . We deduce that q converges, along subsequences as → 0, locally uniformly to q which is Lipschitz continuous. We now show that situations as in Figure 7.1 cannot occur. This is the last step in proving the preliminary heuristic statement that optimal trajectories must look like those in Figure  Lemma 7.4 (The only D-turn occurs at s = 0). -Assume that the conditions of Lemma 7.1 hold. Let θ ∈ R * + and (y, η) ∈ A (α, θ) be an admissible trajectory such that y(s) > α for all s ∈ (s 1 , s 0 ) with s 1 < s 0 < 0 and y(s 0 ) = y(s 1 ) = α. Then (y, η) cannot be an optimal trajectory.
Proof. -We argue by contradiction. Since y(s) > α for all s ∈ (s 1 , s 0 ), it follows that (y, η) satisfies (5.7); however, it remains to determine the matching conditions for p. The fact that s 1 , s 0 < 0 enables us to use Lemma 7.2 to get thatẏ(s 0 ) = y(s 1 ) = 0 and Let us introduce θ 0 = η(s 0 ). Up to a translation in time, we may assume that s 0 = 0 and accordingly,ẏ(0) = 0 and p(0) = α/θ 0 . We shall obtain a contradiction by considering the local convexity of the trajectory y as it comes in contact with the line. During free motion, y is smooth. Since y(0) = α is a local minimum andẏ(0) = 0, it must be that (7.6) lim sup s 0ÿ (s) 0.
We note that we may not conclude thatÿ(0) 0 since we have not established the global C 2 regularity of y. The weaker claim (7.6) does not require this extra smoothness and is sufficient for our purposes. We now use (5.6) on the free portion, s ∈ (s 1 , 0), to collect some identities that we use to contradict (7.6).

The Airy function and related ones
The goal of this subsection is to construct the function Q involved in Proposition 6.2 that plays a key role in establishing Theorem 1.3. Figure 7.2 provides an illustration of Q.
For that purpose, we need to collect some facts about the Airy function Ai, and introduce several auxiliary functions that are useful to prove monotonicity properties of Q. First, we recall that Ai satisfies (7.9) Ai (ξ) = ξ Ai(ξ).
We know the precise asymptotics of Ai as ξ → ∞.
and, for R defined by (1.8), Recall that Ξ 0 is the largest zero of Ai. The asymptotics of R near Ξ 0 are also known. In particular, lim 7.3.1. The auxiliary function E = R Next we introduce one more function. For ξ ∈ (Ξ 0 , ∞), let By the definition of R in (1.8) and by (7.9), we have We summarize further facts in the next lemma.
Next, we prove that E (ξ) < 0 for all ξ. For the sake of contradiction, suppose that there is a critical point, ξ 0 , of E. Then, by (7.14), we have In addition, (7.13) informs us that E(ξ 0 ) = 0. Therefore, ξ 0 is a strict local minimum.
The limiting behavior of E at ∞ implies that there is also a local maximum ξ M ∈ (ξ 0 , ∞). On the other hand, the argument that showed that ξ 0 is a strict local minimum applies to ξ M as well. We conclude that ξ M is both a local maximum and a strict local minimum, which is a contradiction. Hence, there is no critical point, and we have E < 0, and E > 0.

The auxiliary function
We have F(ξ) > 0 for all ξ.

Definition and properties of Q
We are now ready to introduce Q. Lemma 7.7. -The following pair of properties holds true: (i) For each θ > 0, there is a unique solution of q = R (q 2 − θ −1 ) such that q 2 − θ −1 > Ξ 0 . We define the function Q(θ) as the root of this equation. Alternatively speaking, Q(θ) is defined via the following implicit relationship, (ii) The function θ → Q(θ)/θ is strictly increasing, continuously differentiable, and it converges to 1/2 as θ → +∞.

The dynamics on the line
With Lemma 7.4 at hand, we know that trajectories make at most one excursion to the right of the line {y = α}. In the sequel, we show that this excursion occurs if and only if the endpoint (α, θ) is such that θ ∈ (0, θ ), where we refer to the threshold θ given in (6.1), which, according to Lemma 7.7, is uniquely defined. One key step to understanding this is the dynamics on the line. It should be noted, however, that the results in this section are used for more than this one consequence.
In order to prepare the computation of the trajectory off the line (Section 7.5), two constants of integration are needed: A and B, see e.g. (5.7). In the sequel, we gather enough additional equations at the junction with the line in order to resolve the problem. The cornerstone is the relation between q(s ) and η(s ), which is established in Lemma 7.9 below. This enables us to bring the condition at s = −∞ in the definition of A (α, θ) in (4.7) down to a condition at the contact time s = s . Note that the latter is well defined by Lemma 7.1 and Lemma 7.4: Definition 7.8 (Contact time). -Suppose θ > 0, and let (y, η) ∈ A (α, θ) be the optimal trajectory. There exists a unique s = s (θ) 0 such that y(s) = α if and only if s s . Lemma 7.9. -Suppose θ > 0, and let (y, η) ∈ A (α, θ) be the optimal trajectory. Let s ∈ R − be the contact time. Then q and Q α , defined by (5.10) & (6.2), satisfy, for s < s , Proof. -In this proof, we assume that α = 1 without loss of generality. The appropriate relationship (7.19) can be recovered afterwards from the scaling η(s) = α 2/3 η 4/3 (s), see Remark 6.4.
As above, we may also assume without loss of generality that s = 0 up to a time shift of the trajectory. First, we recall that (y, η) ∈ A (α, θ) implies Second, due to Lemma 5.4, we recall that η is non-increasing. In view of (5.10), this implies η 2q. The first conclusion we make from these two facts is that q and η both tend to infinity. Indeed, first suppose that η is bounded. Since η is monotonic, then there exists η ∞ such that η(s) ∈ (θ, η ∞ ) for all s < 0. It follows that q(s) remains bounded from above as well. From (5.10), we find After taking s → −∞, we see that q(s) → ∞, which contradicts the boundedness of η.
Similarly, if q is bounded, there exists q ∞ such that q(s) q ∞ for s < 0. Since η tends to infinity, choose S > 0 large enough so that η(−S) > 2q ∞ . Then, using this as well as (5.10), we obtain, for s < 0, By taking the asymptotic limit, we find: where the second inequality follows from our choice of S. However, this is impossible due to (7.20). Thus, q cannot be uniformly bounded.
In addition, the second equation in (5.10) implies that q is monotonic. We conclude that the following limits hold true, Now, consider the combination of η and q given by It follows from (5.10) thatξ = −1/η. Let φ(s) = R(ξ(s)) − q(s). First notice that, using (7.22), along with (7.10), it follows that φ(s) → 0 as s → −∞. Second, we find, where we have used (7.11) to obtain the second equality. Thus, we obtain, For any s < 0, integrating this from s to 0 yields the identity The definition of ξ in (7.22) and (7.21) imply that lim s → −∞ ξ(s) = +∞. This, together with the asymptotics for R in (7.10), and (7.21), imply that R(ξ(s )) + q(s ) is positive for s < S negative enough (it even tends to infinity). We deduce that The fact that φ(s) → 0 as s → −∞, along with (7.23) and (7.24), imply φ(0) = 0. Using this information, with (7.23) again, shows that φ(s) = 0 for all s < 0. Thus, according to the definition of φ, we have, which is equivalent to q(s) = Q (η(s)). This concludes the proof of Lemma 7.9. We conclude this section with a relatively precise description of the behavior as s → −∞ (or, equivalently, t → 0). Lemma 7.10 (Anomalous behavior as s → −∞). -The following asymptotics hold for optimal trajectories: Note that (3.1), the anomalous scaling in the original variables, follows from Lemma 7.10.
Proof. -We may assume α = 1 up to a scaling argument (6.6), as in the previous proof.
The first step is to show that the contact time is non zero (s < 0) if θ < θ . Alternatively speaking, for endpoints below the threshold, the trajectory makes a free motion excursion in {y > α}.
Proof. -We argue by contradiction. Suppose there exist such times s 1 < s 0 . Then, we test the optimality (y, η) against a perturbation (y + , η) compactly supported in (s 1 , s 0 ), and such that 0 in order to preserve the condition y + α. Then, by the optimality of the trajectory with respect to the Lagrangian (4.6), Since is compactly supported in (s 0 , s 1 ) and since (y, η) satisfies (5.6) on (s 1 , s 0 ), it follows thatη + η = 2q. Hence, By the arbitrariness of 0, it follows that 4q(s) η(s) for all s ∈ (s 0 , s 1 ). However, using Lemma 7.9, this implies that 4Q α (η(s)) η(s), which cannot hold if η(s) < θ by the definition of θ in (6.1) and the monotonicity established in Lemma 7.7.
We set some notation. Given θ 0, let θ (θ) = η(s (θ)) be the value of η at the contact time, where (y, η) is the optimal trajectory associated to (α, θ). We had used this notation already in the proof of Theorem 1.3 (ii).
We continue with a characterization of θ at the contact time. We remark that the map θ → θ , defined on the line {y = α}, connects the two values η(0) and η(s ) at the two extremities of the free excursion.

TOME 5 (2022)
Proof. -Let (α, θ n ) → (α, θ) be a sequence of endpoints, with the associated sequence of optimal trajectories (y n , η n ). Examining the proof of Lemma 7.3 we find a locally uniform H 1 loc bound on both (y n ) and (η n ). By a diagonal extraction argument, we can extract a subsequence such that (y n k , η n k ) converges to some trajectory (y, η) weakly in H 1 loc . Fatou's lemma and the lower semi-continuity of L α enables us to conclude, as in the proof of Lemma 3.1, that Since (y, η) ∈ A (x, θ), then the left hand side is no smaller than U α (α, θ). On the other hand, the convexity of U α implies its continuity and, hence, it implies that lim inf n → ∞ U α (α, θ n ) = U α (α, θ). Taken together, this implies that (y, η) is the minimizing trajectory associated to (α, θ).
We use the rigidity of the expression of the optimal trajectories from (5.7) in order to rule out possible jumps. Indeed, suppose that s 0 < s (θ). Then, passing to the limit on the parameters A n k and B n k (up to another extraction), we get a polynomial function (in the variable τ = e s/2 ) which coincides with α on (s 0 , s ), due to the convergence of y n k to y. This can only happen if A = B = 0. In this case, s (θ) = 0. On the other hand, we find that A n k , B n k → 0, and, thus, s (θ n k ) → 0. This implies that s 0 = 0 = s (θ), which is a contradiction. We conclude that the whole sequence (s (θ n )) converges to s (θ). Therefore, θ → s (θ) is continuous.
The same conclusion holds for θ → η(s (θ)) since η is a continuous function.
As already discussed to motivate the statement in Lemma 7.11, we have s < 0 if the endpoint is such that θ < θ . In fact, the converse is true.
Proof. -We consider α = 1 without loss of generality. To begin with, we collect some useful identities at the time of contact. By definition, we have y(s ) = α and η(s ) = θ . By (5.7), Lemma 7.2, and Lemma 7.9, we have, which, with the usual notation τ = e s /2 , yields, Re-arranging this to obtain an expression for the ratio θ/θ , and then plugging it into (7.26) yields It is a direct consequence of Lemma 7.11 that θ (θ) θ for all θ. Otherwise the optimal trajectory with the endpoint (α, θ ) would stick to the line {y = α} with values η < θ , in contradiction with the statement of the lemma.
Due to the definition of θ in (6.1), this implies that θ 0 = θ , as claimed. Finally, it follows from the optimality of sub-trajectories of the one ending at (α, θ ) that y(s) = α for all s ∈ (−∞, 0] when θ θ . This concludes the proof that s (θ) = 0 for all θ θ .

The complete picture of the trajectories -Proposition 6.2
Proof of Proposition 6.2. -There are a number of items to check.
(i) The existence of the contact time s is a consequence of Lemma 7.1 and Lemma 7.4. The continuity of the map θ → η (s (θ)) is the purpose of Lemma 7.12. (ii) The property η(s (θ)) θ is a consequence of Lemma 7.11. (iii) The fact that s = 0 if and only if θ θ is a consequence of Lemma 7.11 and Lemma 7.13. (iv) We can separate the dynamics on and off the line, respectively for s ∈ (−∞, s ) and (s , 0). On each interval, the Lagrangian is continuous and so the classical theory can be applied. Moreover, (y, η, p, q) is globally continuous provided we define p(s) = α/η(s) for s s , as shown in Lemma 7.2. As a by-product of the classical theory, we have in particular U α (α, θ) = −H α (α, θ, p(0), q(0)).
(v) The derivation of the first integral of motion q(s) = Q α (η(s)) is the purpose of Lemma 7.9. (vi) The formula for A at the contact time is clear (see, e.g., (7.7)). The formula for B follows from the continuity of q along with the matching condition at s = s coming from the combination of (5.7) and Lemma 7.7.

Conclusion and perspectives
We have shown a weak propagation result for the cane toad equation (1.2). More precisely, we have proven that the front spreads slower than the linear problem without saturation. In fact, the linear problem was previously shown to spread as (4/3)t 3/2 , in contrast to the rate α * t 3/2 , where α * ≈ 1.315, obtained here. However, our spreading result is quite weak, and oscillatory behavior could not be ruled out.
Dumont performed intensive numerical computations on a large domain to investigate the long time asymptotics of (1.2). The methods and the results are described in the following appendix. He does not report any oscillatory behavior. The spatial density ρ appears to be monotonic non-increasing with respect to the space variable. In addition, all level lines propagate at the same rate O(t 3/2 ) with the same prefactor. Furthermore, the numerical spatial density converges to a Heaviside function with unit saturated value 1 {x<α h t 3/2 } in the self-similar spatial variable x/t 3/2 , for some numerical critical value α h . This suggests that Theorem 1.2 could be strengthened towards a strong spreading result stating that all level lines propagate as α * t 3/2 . Accordingly, we conjecture that the value function U α, 1 is a good candidate to describe the asymptotic behavior associated with the exponential ansatz discussed in Section 2 as t → ∞. An alternative would be to seek a stationary profile adapted to the various scales of the problem, as discussed in Figure A.

2.
We are not aware of any other reaction-diffusion problem related to the Fisher-KPP equation where the saturation term hinders the propagation at first order. Usually, the non-linear term acts on the next order correction of the front location, as in the Bramson logarithmic delay [BHR17a,Bra83,HNRR13,Pen18]. Our analysis unravels the interplay between unbounded diffusion, curved trajectories due to the twisted Laplacian θ∂ 2 x +∂ 2 θ , and non-local competition among individuals at the same location, but having different dispersal abilities, as shown in Figure 2.2. We believe that the methodology developed here could be extended to other related problems. One clearly sees the joint propagation in (x, θ) towards larger x and higher θ. Moreover, the function U seems to converge to a stationary profile in the selfsimilar variables, in agreement with the heuristic argument of Section 2 that u h → u where u h is given by (2.1), u solves (2.2), and u and U are connected by a change of variables as in Section 4.3.
The numerical approximation of the Cauchy problem (1.2) raises several challenges: (1) Handling the non-local reaction term with an implicit scheme would result in full non-linear systems to be solved at each time step. (2) But as time t increases, diffusion triggers faster and faster time scales as the solution propagates in x. Therefore, an implicit stable discretization seems necessary [HW10].
(3) Experience shows that a large domain and a thin discretization is necessary to achieve good spreading numerical results. Thus, the numerical simulation, whatever the method opted for, requires a large amount of computing time, even with a parallel procedure. A strategy for reducing the computing time appears to be necessary. The spatial density converges to a Heaviside function in the self-similar variable x/t 3/2 , in agreement with the analysis performed in the article. (C) The same curves are plotted, but in the frame centered at the abscissa X 1/2 (t) corresponding to the value ρ = 1/2. Increasing times are figured by an arrow. The front flattens as time increases.
(D) By playing with scales, I found that the typical width of the front is of order t 1/2 , as all curves are superposed in this frame.

A.1. Methods
I opted for standard operator splitting techniques. These techniques date back to the 1950's. However, it has been shown recently that they are well adapted to difficult and even very stiff problems [Des00, DBM + 12]. Being given an initial value and a time step h for the problem df /dt = Lf + R(f ), decomposed into the linear (diffusion) part and the non-linear (reaction) part, I advance from time nh to time (n + 1)h by solving only partial problems: L h : df /dt = Lf and R h : df /dt = R(f ) during the time step h. Denoting by f n the numerical solution at time nh, the Strang scheme [Str68] is of order 2 provided L h/2 and R h are also approximated by numerical schemes of order 2. It can be generalized to three operators [DBM + 12], keeping order 2. Define the three partial problems as: L x h : df /dt = θ∂ 2 f /∂x 2 , L θ h : df /dt = ∂ 2 f /∂θ 2 , and R h : df /dt = R(f ), and the corresponding Strang scheme as follows A.1.1. Numerical computation of each sub-problem I approximated the operators L x h and L θ h using the Crank-Nicolson method, which is of order 2 and A-stable [HW10]. The non-local reaction term R h is non-stiff and was approximated by a second order explicit Runge-Kutta method (RK2) [HNW93].
The spatial discretization was made via second order finite differences, on an uniform grid of size N x × N θ . Notice that L x h and L θ h can be decomposed further in independent sub-problems acting respectively on the rows and on the columns of the finite difference grid. Hence, each sub-problem boils down to solving banded tridiagonal linear systems. Hence the cost of advancing one step in time these two discrete operators reduces to only O(N x × N θ ). Moreover the sub-problems of L x h and L θ h can be computed in parallel. The non-local reaction term R h involves the rate of growth 1 − ρ(t, x) = 1 − ∞ 1 f (t, x, θ)dθ which does not depend on θ. Hence, it can be computed in parallel for each value of x on the grid.
A.1.2. Time step control I used the first order Euler splitting scheme in order to compute I used the quantity f * n+1 − f n+1 L 2 = O(h 2 ) as an error indicator to adapt the time step h, as usually done for solving ordinary differential equations [HNW93].

A.1.3. Implementation
The code was implemented in C++, OpenMP parallel (in shared memory). The size of the domain was (x, θ) ∈ [0, 4.5E4] × [0, 1.6E3]. The size of the grid was N x = 28125 and N θ = 1000 (∆x = ∆θ = 1.6). The time step was adapted at each iteration for small time t, then every 5 steps afterwards. The total wall clock computing time of the simulation was approximately 4 days, on a 32 cores (2.2 Ghz clock frequency) computer.

A.2. Results
Starting from an initial datum as in (1.3), but restricted to the positive values of x, the spatial propagation was observed in the long term with rate O(t 3/2 ). The numerical value of the prefactor seems to converge to a value lying between 1.34 and 1.35 (up to 2.6% of relative error). This discrepancy may be explained by the size of the meshgrid, which is of order one to cope with computing limitations. For the sake of comparison, the same numerical procedure was implemented for the standard Fisher-KPP equation (1.1) with similar numerical parameters (∆x = 1, adaptive time step). The actual wave speed (c = 2) was overestimated by 1.9% of relative error.
The analysis performed in this article suggests that accurate numerical schemes should be developed on the auxiliary function u = −t log f , in the self-similar variables, in order to match with the ansatz in Section 2 and Section 4.3. I checked that the numerical approximation of u = −t log f did converge in self-similar variables to a stationary function ( Figure A.1). This stationary solution is likely to be an approximation of the value function U α . There is indeed a good match (comparison not shown).
To investigate further the consistency of the analysis performed in the article, I checked whether the spatial density ρ(t, x) resembles a Heaviside function µ1 {x<αt 3/2 } . First, I noticed that, despite the lack of maximum principle, the numerical spatial density ρ(t, x) remains below the unit carrying capacity: ρ 1. Moreover, it is monotonic non-increasing in space, and non-decreasing in time, see Figure A.2. The numerical results suggest that the spatial density indeed converges to the Heaviside function 1 {x<α h t 3/2 } , where the critical value α h depends on the numerical approximation of the scheme.
No stationary profile seems to be reached in the long term asymptotics (Figure A.2C). More precisely, the shape of the front flattens as time increases. The typical width appears to be of order O(t 1/2 ), see Figure