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On a model of a population with variable motility

We study a reaction-diffusion equation with a nonlocal reaction term that models a population with variable motility. We establish a global supremum bound for solutions of the equation. We investigate the asymptotic (long-time and long-range) behavior of the population. We perform a certain rescaling and prove that solutions of the rescaled problem converge locally uniformly to zero in a certain region and stay positive (in some sense) in another region. These regions are determined by two viscosity solutions of a related Hamilton-Jacobi equation.


Setting and main results
We study a reaction-diffusion equation with a nonlocal reaction term: xx n + α∂ 2 θθ n + rn(1 − ρ) for (x, θ, t) ∈ R × Θ × (0, ∞), ρ(x, t) = Θ n(x, θ, t) dθ for (x, t) ∈ R × (0, ∞), ∂ θ n(x, θ m , t) = ∂ θ n(x, θ M , t) = 0 for (x, t) ∈ R × (0, ∞), n(x, θ, 0) = n 0 (x, θ) for (x, θ) ∈ R × Θ. This problem, introduced by Bénichou, Calvez, Meunier, and Voituriez [6], models a population structured by a space variable x and a motility trait θ ∈ Θ. Our analysis is focused on the case where the trait space Θ is a bounded subset of R + . The parameters α and r are positive and represent, respectively, the rate of mutation and the strength of the reaction term. The problem (E) is of Fisher-KPP type. The classical Fisher-KPP equation, ∂ t m(x, t) = β∂ 2 xx m(x, t) + rm(x, t)(1 − m(x, t)) for (x, t) ∈ R × (0, ∞), describes the growth and spread of a population structured by a space variable x. The diffusion coefficient β is a positive constant. The behavior of solutions to (F-KPP) has been widely studied, starting from its introduction in [23,28]. The most important difference between (E) and the Fisher-KPP equation is that the reaction term in (E) is nonlocal in the trait variable. This is because competition for resources, which is represented by the reaction term, occurs between individuals of all traits that are present in a certain location. Another key feature of (E) is that the trait θ affects how fast an individual moves -this is why the coefficient of the spacial diffusion in (E) depends on θ. In addition, the trait is subject to mutation, which is modeled by the diffusion term in θ. Thus, (E) describes the interaction between dispersion of a population and the evolution of the motility trait. We further discuss the biological motivation for (E) and review the relevant literature in more detail later on in the introduction.
Main results. Throughout our paper, we assume that the trait space Θ is a bounded interval where θ M and θ m are positive constants. We study classical solutions of (E) with initial condition that is non-negative and "regular enough." We state these assumptions precisely as (A1), (A2) and (A3) in Section 1.1.1.
We study the limit of the n ε as ε → 0. We find that there exists a set on which the n ε converge locally uniformly to zero, and another set on which a certain limit of n ε dθ stays strictly positive. These sets are determined by two viscosity solutions of the Hamilton-Jacobi equation, The function c : R → R arises from the eigenvalue problem (1.10) and is determined by θ m , θ M , r, and α (see Proposition 1.3). One may view the function c as encoding the effect of the motility trait on the limiting behavior of the n ε . Viscosity solutions of (HJ) with infinite initial data play a key role in our analysis and we provide a short appendix where we discuss the relevant known results. For the purposes of the introduction, we state the following lemma: Lemma 1.1. For any Ω ⊂ R, there exists a unique continuous function u Ω that is a viscosity solution of (HJ) in R × (0, ∞) and satisfies the initial condition (1.1) u Ω (x, 0) = 0 for x ∈ Ω −∞ for x ∈ R \Ω.
We are interested in u Ω for two sets Ω determined by the initial data n 0 . We define these two sets, J and K, by J = {x ∈ R : there exists θ ∈ Θ such that n 0 (x, θ) > 0} and K = {x ∈ R : n 0 (x, θ) > 0 for all θ ∈Θ}.
(1. 2) We see that x belongs to J if initially there is at least some individual living at J, and x belongs to K individuals with all traits are present at x. Our main result says that the limiting behavior of the n ε is determined by u J and u K : Let us remark on a special case of Theorem 1.2. Suppose the initial data n 0 is such that the two sets J and K are equal (this occurs if, for example, n 0 is independent of θ). In this case, u J = u K ≤ 0 and so R = {u J < 0} ∪ {u K = 0}, which means that Theorem 1.2 gives information about the limiting behavior of n ε almost everywhere on R, for all times t.
We now explain the biological interpretation of Theorem 1.2. The question is, as time goes on, which territory will be occupied by the species and which will be left empty? To answer this question, it is enough to determine where the functions u J and u K are zero. In fact, we have the following corollary, which gives information about the limit of the n ε simply in terms of the sets J, K and the constants r, θ m and θ M . For simplicity we consider only the case that J and K are intervals. Corollary 1.1. Assume (A1), (A2) and (A3). Let us also suppose that each of J and K is a (possibly infinite) interval.
Thus, we see that at time t, if we stand a point that is "far" from J, then there are no individuals at x. On the other hand, if we stand at a point x that is "pretty close" to K, then there are some individuals living at x.
We remind the reader of the fact, due to Aronson and Weinberger [4], that 2 β r is the asymptotic speed of propagation of fronts for (F-KPP). Thus, we may interpret Corollary 1.1 as saying that, in the limit, the population we're considering spreads slower than one with constant motility θ M and faster than one with constant motility θ m .
We further discuss the biological interpretation of our results, and give the proof of Corollary 1.1, in Section 1.2.
Biological motivation. Biologists are interested in the interplay between traits present in a species and how the species interacts with it environment -in other words, between evolution and ecology [37,36,35,27]. It has been observed, for example in butterflies in Britain [39], that an expansion of the territory that a species occupies may coincide with changes in a certain trait -the butterflies that spread to new territory were able to lay eggs on a larger variety of plants than the butterflies in previous generations.
Some biologists have focused specifically on the interaction between ecology and traits that affect motility. Phillips et al [34] recently discovered a species of cane toads whose territory has, over the past 70 years, spread with a speed that increases in time. This is very interesting because this is contrary to what is predicted by the Fisher-KPP equation [23,28,4] and has previously been observed emperically [38]. Spacial sorting was also observed -the toads that arrive first in the new areas have longer legs than those in the areas that have been occupied for a long time. In addition, it was discovered that toads with longer legs are able to travel further than toads with shorter legs. It is hypothesised that the presence of this traitlength of legs -is responsible for both the front acceleration and the spacial sorting. Similar phenomena were observed in crickets in Britain over a shorter time period [39]. In that case, the motility trait was wingspan.
The cases we describe demonstrate the need to understand the influence of a trait -in particular, a motility trait -on the dynamics of a population.
Literature review. The Fisher-KPP equation has been extensively studied, and we refer the reader to [23,28,3,22,33] for an introduction. Hamilton-Jacobi equations similar to (HJ) are known to arise in the analysis of the long-time and long-range behavior of (F-KPP), other reaction-diffusion PDE, and systems of such equations -see, for example, Evans and Souganidis [20] and Barles, Evans and Souganidis [5] and Fleming and Souganidis [24]. The methods of [20,5,24] are a key part of our analysis of (E ε ).
As we previously mentioned, (F-KPP) describes populations structured by space alone, while there is a need to study the interaction of dispersion and phenotypical traits (in particular, motility traits). Most models of populations structured by space and trait either consider a trait that does not affect motility or do not consider the effect of mutations. Champagnat and Méléard [14] start with an individual-based model of the such a population and derive a PDE that describes its dynamics. In the case that the trait affects only the growth rate and not the motility, Alfaro, Coville and Raoul [1] study this PDE, which is a reaction-diffusion equation with constant diffusion coefficient: The population modeled by (1.3) has a preferred trait that varies in space. Berestycki, Jin and Silvestere [8] analyze an equation similar to (1.3), but with a different kernel K and growth term r that represent the existence of a trait that is favorable for all individuals. The aims and methods of [1,8] are quite different from those in this paper. The main result of [1] is the existence of traveling wave solutions of (1.3) for speeds above a critical threshold. In [8], the authors establish the existence and uniqueness of travelling wave solutions and prove an asymptotic speed of propogation result for the equation that they consider. We also mention that a local version of (1.3) was investigated by Berestycki and Chapuisat [7]. A model in which the dispersal rate does depend on the trait, but the trait is not subject to mutation, was studied by Desvillettes, Ferriere and Prévost [17] and Arnold, Desvillettes and Prévost [2]. There has also been analysis of traveling waves and steady states for equations of the form where the reaction term v * φ is the convolution of v with some kernel φ. We refer the reader to Berestycki, Nadin, Perthame and Ryzhik [9], Hamel and Ryzhik [25] and the references therein. An important difference between (1.4) and (E) is that the reaction term of (E) is local in the space variable and nonlocal in the trait variable, while the reaction term in (1.4) is fully nonlocal. The long time behavior of solutions to (1.4) is studied in [25]. In addition, [25, Theorem 1.2] establishes a supremum bound for solutions of (1.4). Bouin and Mirrahimi [12] analyze the reaction-diffusion equation with Neumann conditions on v on the boundary of Θ. The main difference between (1.5) and (E) is that the coefficient of spacial diffusion in (1.5) is constant, which means that (1.5) models a population where the trait does not affect motility. The methods used in [12] to analyze (1.5) are similar to those of [20,5,24] and to some of those in our paper. However, in general it is easier to obtain certain bounds for solutions of (1.5) than for solutions of (E). For example, because the coefficient of ∆v in (1.5) is constant, integrating (1.5) in θ implies that Θ v(x, θ, t) dθ satisfies a local equation in x and t that enjoys the maximum principle. This immediately implies that Θ v(x, θ, t) dθ is globally bounded [12,Lemma 2]. This strategy does not work for (E). Indeed, a serious challenge in studying (E), as opposed to (F-KPP) or nonlocal reaction diffusion equations with constant diffusion coeffiecient such as (1.5), is obtaining a global supremum bound for solutions of (E). Let us discuss the literature that directly concerns (E) and (E ε ). The problem (E) was introduced in [6]. The rescaling leading to (E ε ) was suggested by Bouin, Calvez, Meunier, Mirrahimi, Perthame, Raoul, and Voituriez in [11]. In addition, formal results about the asymptotic behavior of solutions to (E) were obtained in [11]. In particular, Part (I) of Proposition 1.2 of our paper was predicted in [11, Section 2]. Bouin and Calvez [10] also study (E). They prove that there exist traveling wave solutions to (E) but do not analyze whether solutions converge to a traveling wave. In fact, to our knowledge, there are no previous rigorous results about the asymptotic behavior of solutions of (E) or the limiting behavior of solutions to (E ε ). The main difficulty is the lack of comparison principle for (E).

Contribution of our work.
• To the best of our knowledge, Theorem 1.1 is the first global supremum bound for a Fisher-KPP type equation with a nonlocal reaction term and non-constant diffusion. • Theorem 1.2 completes the program that was proposed in [11] for analyzing the asymptotic behavior of the model (E) in the case where the trait space Θ is bounded. • We view our main result, Theorem 1.2, as evidence that the presence of a motility trait does affect the limiting behavior of populations. We discuss this in subsection 1.2. • We hope that our work is a step towards analyzing (E) in the case where the trait space Θ is unbounded. It is in this case that the phenomena of accelerating fronts is predicted to occur [11].
Elements of the proofs of the main results. The proof of Theorem 1.1 is quite involved. The difficulty comes from the combination of the nonlocal reaction term and non-constant diffusion. Our proof of Theorem 1.1 uses regularity estimates for solutions of elliptic PDE, a heat kernel estimate, and an averaging technique similar to that of [25,Theorem 1.2]. We believe this combination of methods is new and may be useful in other contexts. We include a detailed outline in Subsection 2.1.
We use a perturbed test function argument (Evans [19]) and techniques similar to the proofs of [5, Propositions 3.1 and 3.2] to establish: Proposition 1.2. Assume (A1), (A2) and (A3) and letū and u be given by (1.7). Then: (I):ū is a viscosity subsolution and u is a viscosity supersolution of (HJ) in R × (0, ∞); and (II): we have 3. An interesting question is whether Theorem 1.2 can be refined to obtain better information about the limit of the ρ ε in the interior of the set {u K = 0}. For instance, is lim inf * ρ ε bounded from below in this set?
Structure of our paper. In the next subsection, we state our assumptions, give notation, and state a certain known result about the spectral problem that defines the function c. In subsection 1.2, we discuss the interpretation of Theorem 1.2 and give the proof of Corollary 1.1. The rather lengthy Section 2 is devoted to the proof of Theorem 1.1. This section is self-contained. In Section 3 we prove Proposition 1.1. The proof of Proposition 1.2 is is Section 4. The proof of Theorem 1.2 are given in Section 5. We also provide an appendix with a discussion of results on comparison (Proposition A.1) and existence and uniqueness (Lemma 1.1) for Hamilton-Jacobi equations with infinite initial data.
We have organized our paper so that a reader who is interested mainly in our proof of Theorem 1.1 may only read Section 2. On the other hand, a reader who is interested in our results about the limit of the n ε , and not in the proofs of the supremum bound on n ε and the estimates on u ε , may skip ahead to Sections 4 and 5 after finishing the introduction.

1.1.2.
Notation. We will slightly abuse notation in the following way. If Q is a subset of R × (0, ∞), then we will use Q × Θ to denote the set of (x, θ, t) such that (x, t) ∈ Q and θ ∈ Θ. We record this as: : (x, t) ∈ Q and θ ∈ Θ}.
Moreover, we have that the map λ → c(λ) is continuous, and satisfies, for all λ,

Interpretation of Theorem 1.2.
Informal discussion. According to [20,Theorem 1.1], the behavior of solutions to (F-KPP) is characterized by a solution v of the Hamilton-Jacobi equation (We remark that this is the negative of the equation that appears in [20]; we write it this way to be consistent with the signs employed in the rest of this paper.) Let us compare (1.11) and (HJ). We see that both are of the form max{v t − H(v x ), v}, but for different Hamiltonians H. It is this difference that captures the effect of the trait. Indeed, according to Proposition 1.3, the term u x c(u x ) satisfies Thus, we see that the term u x c(u x ) in (HJ) is "like a quadratic", but of different size than the quadratic in (1.11). In addition, in [10, Proposition 5] the function −λ c(λ) is characterized as where γ(λ) is positive for all λ. Hence the Hamiltonian in (HJ) is never exactly a quadratic, and hence must be different from that of (1.11). Since (HJ) and (1.11) characterize the behavior of populations with and without a motility trait, respectively, we interpret our results as evidence that the presence of the trait does affect the asymptotic behavior of a population. We make these ideas more precise in the proof of Corollary 1.1.
Proof of Corollary 1.1. Corollary 1.1 follows from Theorem 1.2 and the following lemma.
Proof of Lemma 1.2. The proof of the first statement a direct modification of [32,Lemma 1.5], where this result is established in the case Ω is a single point. The expression for the zero set of v follows easily. We omit the details.
Proof of Corollary 1.1. Let us denote byv the solution to (1.11) with β = θ M and initial data (1.1) with Ω = J. And, let us denote by v the solution to (1.11) with β = θ m and initial data (1.1) with Ω = K. Finally, we take u J and u K to be as in Theorem 1.2. At time t = 0 we have u J =v and u K = v. The estimates (1.12) on −p c(p) imply that u J is a subsolution of the equation forv and u K is a supersolution of the equation for v. Since (1.11) satisfies the comparison principle (see [5,Section 4] or our discussion in the Appendix), we obtain, for all ( We have that u J and u K are non-positive. Therefore, ifv( We now combine this with the conclusions of Theorem 1.2: • Ifv(x, t) < 0, then lim ε→0 n ε (x, θ, t) = 0 for all θ ∈ Θ.
• If (x, t) is in the interior of {v = 0}, then lim sup Using the exact expressions for the zero sets ofv and v given by Lemma 1.2 concludes the proof.

Supremum bound
This section is devoted to the proof of: and satisfies (E) in the classical sense. Assume that Θ satisfies (A3) and the initial condition n 0 satisfies (A2). There exists a constant C that depends only on θ m , θ M , α, r and η such that .
We briefly explain notation for norms and seminorms, which we will be using only in this section. For U ⊂ R d+1 we denote: We present the supremum bound for solutions of (E ε ): On the left is a cartoon graph of the extension θ → a(θ). On the right is a cartoon graph of the extension θ →n(x, θ, t). The parts of the graphs that are in black and very thick represent the original functions θ → θ and θ → n(x, θ, t) on the domain Θ. 1 Corollary 2.1. Assume (A1), (A2) and (A3). There exists a constant C that depends only on θ m , θ M , α, r, η, and |n 0 (x, a(θ))| 2+η,R×R such that for all ε > 0, Let us explain how Corollary 2.1 follows from Theorem 2.1.
Please see Figure 1.
We use a to extend θ → n(x, θ, t) to a functionn(x, θ, t) = n(x, a(θ), t) defined for all θ ∈ R. This is equivalent to extending θ → n(x, θ, t) by even reflection across θ = θ M (since n satisfies Neumann boundary conditions, this gives a periodic function on (θ m , θ m + 2(θ M − θ m )) with derivative zero on the boundary) and then by periodicity to the rest of R. Please see Figure 1.
If we use a to extend an arbitrary twice differentiable function on Θ with derivative zero at the boundary to all of R, the result would still have continuous first derivative but would not necessarily be twice differentiable in θ. However, it turns out that since n solves the equation (E), the extensionn is in C 2,η . Moreover, the Hölder norm of the second derivatives ofn is bounded in terms of the supremum of n. We state this precisely in the following Proposition.
Proposition 2.1 follows from standard estimates and interpolation inequalities for seminorms.
Instead of working directly withn, we work with its averages in θ on small intervals: we define the function v by Our idea to use these averages came from reading the proof of [25,Theorem 1.2] for all x ∈ R, ζ ∈ R, and t ∈ (0, ∞).
The first step in our proof of the supremum bound on n is to obtain a bound on sup x,ζ,t v(x, ζ, t) for some σ > 0. We then use the following proposition to extend the bound on v to the supremum bound on n itself.
There exists a constant C 0 that depends only on α, θ m and θ M such that, for all x and θ, We will use the observation that n(x, θ, t)e −t is a subsolution of the heat equation and apply this Proposition with initial data n(x, θ, t * ) at a certain fixed time t * . Proposition 2.2 follows from certain estimates for the heat kernel associated to the operator ∂ t − a(θ)∂ 2 xx − ∂ 2 θθ . These estimates say that this kernel is like the kernel for the heat equation and were established by Aronson [3].
We postpone the proofs of the propositions and use them to present:

Proof of the supremum bound.
Proof of Theorem 2.1. Let C 0 be the constant from Proposition 2.2 and letC be the constant from Proposition 2.1. Define the constantsC and M 0 by We claim If (2.7) holds, then we will have sup which will complete the proof. We proceed by contradiction: let us assume M is a number with M > M 0 and Throughout the rest of the proof of this proposition, C denotes a positive constant that may change from line to line and depends only on θ m , θ M , r, α and η (in particular, C does not depend on M ).
Let us consider the map S(t) that takes t to the supremum of n at time t, in other words: The map S is continuous. In addition, since n satisfies (E), and n and ρ are non-negative, we have that n(x, θ, t)e −t is a subsolution of (2.9) The equation (2.9) satisfies the comparison principle. Therefore, we have the following bound on n(x, θ, t)e −t from above: for all x ∈ R and θ ∈ Θ, Taking supremum in x and θ and multiplying by e t gives the bound for all t > 0. In particular, taking supremum over t ∈ (0, 1], we have We will now work with the extensionn(x, θ, t) defined in Proposition 2.1. By the previous line, we have We apply Proposition 2.1 ton. Part 1 implies thatn satisfies equation (2.3). Part (2) gives us the estimate where the second inequality follows since M ≥ 1. We use our choice ofC in (2.5) to bound the right-hand side from the previous line from above and obtain,

First
Step: We will prove sup Since n satisfies (E), we have that v satisfies Let us explain why we may assume, without loss of generality, that the supremum of v on R × Θ × (0, T ) is achieved at some (x 0 , ζ 0 , t 0 ). Since ζ → v(x, ζ −θ m , t) is periodic of period 2(θ M −θ m ), and we are considering times t in the bounded interval (0, T ), we know that there exist ζ 0 ∈ Θ, t 0 ∈ [0, T ] and a sequence {x For each k, we define the translated functions We similarly define the translates of the first and second derivatives of v. We haven xx ∈ C η , and according to (2.11), Therefore,n k xx , ρ k , v k , and the translates of the first and second derivatives of v are uniformly bounded and uniformly equicontinuous on R × R × [0, T ]. Hence, there exists a subsequence (still denoted by k) and functions ρ ∞ ,n ∞ xx , and v ∞ such that ρ k ,n k xx and v k converge locally uniformly to ρ ∞ ,n ∞ xx and v ∞ , respectively; the derivatives of v k converge locally uniformly to those of v ∞ ; and v ∞ (x, ζ, t) = We point out that v xx does not appear in (2.12), the equation that v satisfies. We will bound from above the corresponding term that does appear in (2.12). This term is: The inequality (2.14) implies that there exists θ * ∈ (ζ 0 − σ/2, Since a is positive, we multiply (2.14) by −a(θ) and find Adding the term (2.15) that we're interested in to both sides of this inequality, we find We thus use the estimate (2.11) on the seminorm ofn xx and the fact thatn xx (x 0 , θ * , t 0 ) ≤ 0 to estimate the right-hand side of the previous line from above and obtain Since θ * ∈ (ζ 0 − σ/2, ζ 0 + σ/2), we have |θ − θ * | ≤ σ. In addition, a is Lipschitz with Lipschitz constant 1, so we have (a(θ) − a(θ)) ≤ |θ −θ|. We use these two inequalities to bound the right-hand side of the previous line from above and find The last inequality follows by an elementary calculus computation that relies on the fact thatθ is contained in Using this estimate together with the information (2.13) about the other derivatives of v at (x 0 , ζ 0 , t 0 ) in the equation (2.12) that v satisfies, we obtain We use the previous estimate to bound the right-hand side of (2.16) from above and obtain Upon rearranging we find, . We use this to bound the right-hand side of the previous line and find Since v achieved its supremum on R × R × (0, T ) at (x 0 , ζ 0 , t 0 ), we conclude

Second
Step: We will now deduce a supremum bound onn from the bound on v. Let us fix any t * ∈ (1, T ). Let u be the solution of We have thatn(x, θ, t + t * − 1)e −t is a subsolution of the equation for u for t ≥ 0. Since they are equal at t = 0, the comparison principle for the equation for u implies the bound for all x, θ, and t ≥ 0. In particular, we evaluate the above at t = 1 and take supremum in x and θ to find  u(x, θ, 1).
We will now apply Proposition 2.2 to u. The supremum bound (2.17) on v says exactly that assumption (2.4) is satisfied, with C 1 = 3. Therefore, Proposition 2.2 implies sup x∈R,θ∈R Therefore, we may bound u by 3C0 σ on the right-hand side of (2.18) and find, This holds for any t * ≤ T , so in particular at t * = T . According to line (2.10), we have sup x,θn (x, θ, T ) = M , so we obtain which implies, since 1 − 4+η 4+2η = η 4+2η , the following bound for M : Therefore, But we had taken M > M 0 . Recalling our choice of M 0 in line (2.6) yields the desired contradiction and hence the proof is complete.

Proofs of auxillary results for the supremum bound.
Proof of Proposition 2.2. The proof of Proposition 2.2 relies on a certain L ∞ estimates on the heat kernel, which we formulate as the following lemma.
Lemma 2.1. There exists a kernel K such that ifū 0 is non-negative andū is a weak solution of x, y, θ, r)ū 0 (y, r) dr dy.
Moreover, there exist constants c 1 and c 2 that depend only on α, θ m and θ M such that for all x, y, θ in R, and for all t > 0.
Proof. We apply two theorems of Aronson [3]. That we haveū = K * ū 0 is the content of [3, Theorem 11]. The bound on K is exactly [3, item (ii), Theorem 10]. Aronson's results apply to any parabolic equation with bounded coefficients and with G regular enough. In our case, n = 2 and G ≡ 0 (so, in particular, it satisfies Aronson's hypotheses) and A(x, θ) is the diagonal matrix with entries a(θ) and α. In the general case, the constants c 1 and c 2 depend on the ellipticity constant of A and the L ∞ norm of A. Since these depend only on α, θ m and θ M , we have that c 1 and c 2 depend only on α, θ m and θ M as well.
For the proof of Proposition 2.2 we also need the following lemma. Its proof is elementary enough to give as an exercise in a calculus class.
Proof. We have a bound on the series we're interested in by the corresponding integral of e −a 2 x 2 : By change of variables, we obtain We proceed with: Proof of Proposition 2.2. According to Lemma 2.1, we have, for all x ∈ R, all θ ∈ R, and all t > 0, Let us take t = 1 to obtain the following bound forū(x, θ, 1): (2.19) We will now split up the integral in r into a sum of integrals over intervals of size σ. We have For r ∈ (iσ, (i + 1)σ) we have e −cr 2 ≤ e −ci 2 σ 2 .
We use this to bound each of the integrals in r and find, For each i, we can take ζ = i − σ/2 in assumption (2.4) of this proposition to obtain where the last inequality follows from the elementary bound on the sum of the above convergent series of Lemma 2.2. Since σ ≤ 1 we find We now use this bound on the integral in r in the estimate (2.19) forū(x, θ, 1) and obtain This holds for all x and θ, so the proof is complete.
Proof of Proposition 2.1. We use the following result on the solvability in C 2,η of parabolic equations: Proposition 2.3. Let us take η ∈ (0, 1) and suppose the coefficients a ij are uniformly elliptic, a ij ∈ C η (R 2 ), f ∈ C η (R 2 × (0, T )), and u 0 ∈ C 2+η (R 2 ). Then the initial value problem has a unique solution u ∈ C 2+η (R 2 × [0, T )). Moreover, there exists a constant C that depends on η, the ellipticity constant of a ij , and |a ij | η,R 2 such that Proof. That there exists a unique solution u ∈ C 2,η is part of the statement of Ladyzenskaja et al [ where the constant C 1 depends only on η, the ellipticity constant of a ij and |a ij | η,R 2 . We use the equation that u satisfies to bound from above the first term on the right-hand side of (2.21) and find, Using the previous line to bound from above the first term on the right-hand of (2.21) yields the estimate (2.20).
In addition, we need the following lemma: Lemma 2.3. Assume Θ satisfies (A3). Given T > 0 and continuous and non-negative functions ρ(x, t) and u 0 (x, θ), the solution u of We point out that here ρ is a given function and the equation (2.22) for u is local. Because of this, the proof of the lemma is standard and we omit it.

Estimates on u ε
In this section we give the proof of Proposition 1.1. We believe that the arguments in this section are the most technical ones of the paper.
Let us remark that u ε (x, θ, t) = ε ln n ε (x, θ, t) satisfies In subsection 3.1 we establish the lower bound on u ε by a barrier argument that is quite similar to [ ). This argument is quite lengthy and has a some rather technical parts. We provide all the details.
3.1. Proof of lower bound of Proposition 1.1. In order to construct the barrier necessary for the proof of the lower bound, we need a lemma. The proof of the lemma is an elementary computation which we include for the sake of completeness.
Lemma 3.1. Given R < 1/2 we define the function ξ(z) by, There exists a constant C that depends only on R such that for all z ∈ (−R, R) and for all 0 < ε < 1, Proof. Computing the derivatives of ξ, substituting them into −ξ zz (z) − (ξ z (z)) 2 and rearranging yields, for z ∈ (−R, R), We have R 2 − z 2 ≤ R 2 ≤ 1/4, so that the term in the inner-most parentheses on the right-hand side of the previous line is bounded from above by −2. Consequently we find, for z ∈ (−R, R), where we have again used that |z| ≥ R √ 2 . Hence the term in the parenthesis on the right-hand side of (3.1) is bounded from above by −3. Therefore, the right-hand side of (3.1) is negative for R √ 2 ≤ |z| ≤ R. Thus it is left to consider z such that |z| ≤ R √ 2 . But in this case, we see R 2 ≥ R 2 − z 2 ≥ R 2 /2. Using this, together with the fact that the term in the parenthesis on the right-hand side of (3.1) is bounded from above by 1, and find, Since ξ zz (z) is non-positive for z ∈ (−R, R), we have, for 0 < ε < 1 and z ∈ (−R, R),

This completes the proof of the lemma.
Proof of the lower bound of Proposition 1.1 . Due to Corollary 2.1, there exists a positive constant M that depends only on θ m , θ M , α and r such that u ε is a supersolution of Without loss of generality we may take R < 1/2. Since n 0 is continuous and positive inK R , there exists a constant β ≤ 1 such that infK R n 0 ≥ β. Therefore, if (x, θ) is contained inK R , we have u ε (x, θ, 0) = ε ln n 0 (x, θ) ≥ ln(β).

3.2.
Proof of the gradient bound of Proposition 1.1. Most of this subsection is devoted to the proof of: where the coefficients A and H are given by Given Q ⊂⊂ Q ⊂⊂ R × (0, T ), there exists a constant C that depends only on Q, Q , ||u|| L ∞ (Q ×Θ) , θ m , θ M , α, r and M such that Proposition 3.1 is essentially the same as Part (II) of Proposition 1.1, but we believe the notation in Proposition 3.1 is more suited to the presentation of the proof. Let us now describe how Part (II) of Proposition 1.1 follows from Proposition 3.1.
Proof of Part (II) of Proposition 1.1. We apply Proposition 3.1 to u ε with f = r(1 − ρ ε ). According to Part (I) of Proposition 1.1, which we have just established, we have that u ε is bounded on Q × Θ, uniformly in ε.
According to Corollary 2.1 we have that n ε , and hence ρ ε , is uniformly bounded from above. In addition, ρ ε is non-negative. Hence the hypothesis (3.6) is satisfied. Now let us demonstrate that (3.6) holds. Let us define n(x, θ, t) = n ε (εx, θ, εt). Since n ε satisfies (E ε ) and n 0 satisfies (A2), we have that n satisfies (E) with initial data that satisfies (A2). Thus, according to Proposition 2.1 and Theorem 2.1, n is bounded in C 2,η . Therefore, there exists a constant C so that, for all (x, θ, t), Integrating in θ yields the desired estimate for ρ ε x . For the proof of Proposition 3.1 we will need the following elementary lemma.
Lemma 3.2. For any diagonal n × n positive definite matrix M , any n × n matrices X, Y , and any β > 0, we have In addition, if X is symmetric then tr(M XM X) = tr(M M XX).
Proof. Let us fix M and define ·, · by X, Y = tr(M XY T ). We have that ·, · is an inner product on the space of n × n matrices. Hence, ·, · satisfies the Cauchy Schwartz inequality. In other words, we have X, Y 2 ≤ X, X Y, Y for all n × n matrices X and Y . Taking square root and using Cauchy Schwartz for real numbers gives, Using the definition of , gives that (3.9) holds. For any three symmetric matrices A, B, C, we have tr(ABC) = tr(BAC). We apply this with A = M X, B = M and C = X and obtain the second claim of the lemma.
Proof of Proposition 3.1. Let ψ(y 1 , t) be a cutoff function supported on Q and identically 1 inside Q. Let us define ζ(y, t) = ψ 2 (y 1 , t) andH (p, y, t) = ζ(y, t)H(p, y). We record for future use the following facts: (3.10) H p (Du, y) · Du = 2H(Du, y) where we denote W = 0 u y1 0 0 . Throughout the remainder of the argument we will use C and C i with i = 1, 2, ... to denote constants that may depend on α, r, ||u|| L ∞ (Q ) , θ m , θ M , Q and M . In addition, C may change from line to line. We also define the constant λ by where the constants C i are specified later in the argument. We define the function (3.12) z(y, t) =H(Du, y, t) + λu(y, t) and we claim (3.13) if (y 0 , t 0 ) is the maximum of z on Q × Θ and (y 0 , t 0 ) / ∈ (∂Q ) × Θ, then H(Du(y 0 , t 0 ), y 0 ) ≤ C.
We use thatH = ζH ≤ H and the estimate (3.16) to bound the first term on the right-hand side of the previous line of the above by C, and that ||u|| L ∞ (Q ×Θ) ≤ C to bound the last two terms. This yields (3.14).
Second step: Next we will use (3.14) to verify the following upper bound on u 2 y2 in terms of C and λ: To this end, we use the definition ofH and the fact that ζ is identically 1 inside Q to obtain, α ε 2 u 2 y2 (y, t) =H(Du, y, t) − y 2 u 2 y1 for all (y, t) ∈ Q × Θ.
We use the estimate (3.14) to bound the first term on the right-hand side of the previous line from above by C + λC. Since the second term is negative, we obtain the estimate (3.17). Third step: There are two possible values for λ, and we will show that, together with the estimate (3.17) that we just established, either one implies the desired estimate (3.8). Let us first suppose λ = Cε −1 . Upon substituting this on the right-hand side of (3.17) we find, which yields the desired estimate (3.8) by multiplying both sides by ε 2 . Now let us suppose that λ takes on the other possible value, so that λ = C sup Q (ζu y1 ) . We use that θ m ≤ y 2 , the definition ofH, the line (3.14) and the value of λ to obtain, for any (y, t) ∈ Q × Θ, From this we deduce sup Q (|ζu y1 |) ≤ C and hence λ ≤ C. Using this on the right-hand side of (3.17) implies that (3.8) holds. The proof of (3.13). Let us suppose (y 0 , t 0 ) is the maximum of z on Q × Θ and (y 0 , t 0 ) / ∈ (∂Q ) × Θ, so that we have (3.18) 0 ≤ z t (y 0 , t 0 ) − tr(AD 2 yy z(y 0 , t 0 )), and D y z(y 0 , t 0 ) = 0. (If (y 0 , t 0 ) is in the interior of Q × Θ, then of course (3.18) holds. Otherwise, we have (y 0 , t 0 ) ∈ Q × ∂Θ, and the Neumann conditions for u on ∂Θ imply that (3.18) holds.) We seek to establish (3.19) H(Du(y 0 , t 0 ), y 0 ) ≤ C.
As we have just shown, once we establish (3.19) the proof of the proposition will be complete. To this end, we compute, (3.20) z t − tr(AD 2 yy z) = λ(u t − tr(AD 2 u)) + ζH p (Du, y) · (Du t − tr(AD 3 u)) + I where I is the sum of the left-over terms from z t and − tr(AD 2 z): I =H t (Du, y, t) − tr A H pp (Du, y, t)D 2 uD 2 u + 2H py (Du, y, t)D 2 u +H yy (Du, y, t) .
(Throughout, we use D to denote the derivative in y.) We use that u satisfies (3.5) to write the first term on the right-hand side of (3.20) as (3.21) λ(u t − tr(AD 2 u)) = λ(H(Du, y) + f (y)) ≤ λ(H(Du, y) + r), where we have used (3.6) to obtain the inequality. Now let us look at the second term on the righthand side of (3.20). We recognize that Du t − tr(AD 3 u) is "almost" the derivative of u t − tr(AD 2 u), up to a term that involves a derivative of A. In addition, rearranging the equation that u solves implies u t − tr(AD 2 u) = H(Du, y) + f (y). We find: Du t − tr(AD 3 u) = D(u t − tr(AD 2 u)) + tr(DAD 2 u) = D(H(Du, y)) + Df (y) + tr(DAD 2 u).
We now claim that the sum of the leftover terms I and II is bounded: We point out that once the bound (3.26) is established, the proof of (3.19), and hence of the proposition, will be complete. Indeed, using (3.26) to estimate the right-hand side of (3.25) yields 0 ≤ − λ 2 H(Du(y 0 , t 0 ), y 0 ) + Cλ, which, upon rearranging and dividing by λ > 0 yields (3.19).
Proof of bound (3.26) on error terms. Let us start with I. We will prove, We use the expressions (3.11) forH pp andH py to rewrite I as Let us bound from above the third term in I by applying the inequality of Lemma 3.2 with M = A 2 , X = Du ⊗ Dψ, Y = ψD 2 u and β = 2. We obtain: Let us apply Lemma 3.2 again, this time with M = Id, X = W , Y = AD 2 u and β = 2. We obtain the following upper bound for the fourth term in I: where we have also used the second statement in Lemma 3.2. We use (3.29) and the previous line to bound from above the third and fourth terms, respectively, in on the right-hand side of (3.28). Notice that the terms involving A 2 D 2 uD 2 u in the previous line and in (3.29) will be "absorbed" by the second term in I (we are using that ψ 2 = ζ). We obtain: The first term on the right-hand side of (3.30) is simply ζ t H(Du, y), which is less than λ 16 H(Du, y). In addition, we have 8ζu 2 y1 ≤ C 1 H(Du, y) ≤ λ 16 H(Du, y). We use this to bound the right-hand side of (3.30) from above and find, We will show now show that each of the last two terms on the right-hand side of the previous line is less than λ 16 H(Du, y). Once we show this, the estimate (3.27) will be established. To this end, we use that ψ is independent of y 2 , to compute Multiplying by 8/ε and using the definitions of H and λ we obtain, Let us estimate the last term of (3.31). We computeH yy in terms of the derivatives of H and ζ and find, We multiply both sides of the previous line by −A(y) and take trace. The first term gives zero. The second is simply −H(Du, y)ζ y2y2 α ε . We summarize this as, − tr(AH yy (Du, y)) ≤ C 4 ε −1 H(Du, y) ≤ λ 16 H(Du, y).
Since since Dζ = (ζ y1 , 0), the first term in II is simply For the second term in II, we use that Df = (f y1 , 0), the Cauchy-Schwartz inequality, the assumption (3.7), and the definitions of H and λ to find, Finally we will bound the last term in II. We have D y1 A ≡ 0, so the last term is simply where the first inequality follows by applying Cauchy Schwartz and the second from the definitions of A and H. We therefore find, Adding our upper bound (3.27) on I to the previous line we obtain, as desired. Thus the proof of Proposition 3.1 is complete.

Limits of the u ε as ε approaches zero
This section is devoted to studying the half-relaxed limitsū and u of the u ε that we mentioned in the introduction. For the convenience of the reader, we recall their definitions here: u(x, t) = lim ε→0 inf{u ε (y, θ, s) : ε ≤ ε, |y − x|, |t − s| ≤ ε, θ ∈ Θ}.
We postpone the proof of the lemma until the end of subsection 4.1.

4.1.2.
Proof that u is a supersolution of (HJ). We place one computation into a separate lemma.
is a smooth function that satisfies, for some (x 0 , t 0 ), We set λ = ∂ x v(x 0 , t 0 ) and take Q(θ) to be the solution of the spectral problem (1.10) corresponding to this λ. We define There exist positive constants s, ε 1 such that for all ( We postpone the proof of the lemma and proceed with: Proof that u is a supersolution of (HJ). Let us fix a point (x 0 , t 0 ) ∈ R × (0, ∞) such that u(x 0 , t 0 ) < 0. By Proposition 1.1, u(x 0 , t 0 ) is finite, so we denote −δ = u(x 0 , t 0 ). We aim to prove that u satisfies (HJ) in the viscosity sense, so let us suppose u − v has a local minimum at (x 0 , t 0 ) for some smooth function v. We shall show We proceed by contradiction and assume that (4.5) does not hold. Therefore, there exists a > 0 such that v satisfies (4.2). We define the perturbed test function v ε (x, θ, t) by (4.3). According to Lemma 4.2, there exist s > 0 and ε 1 > 0 such that v ε satisfies (4.4) on B s (x 0 , t 0 ) × Θ for all ε ≤ ε 1 . By Proposition 1.1, there exists some positive constants c, C, ε 0 such that for all ε < ε 0 and for all (x, t, θ) ∈ B c/2 (x 0 , t 0 ) × Θ, (4.6) |u ε θ (x, θ, t)| ≤ ε 1/2 C. Let (x εn , θ εn , t εn ) be the sequence of points given by Lemma 4.1. According to item (3) of Lemma 4.1, we have lim Therefore, for all n large enough, we find, Together with the estimate (4.6) on ∂ θ u ε , this implies that there exists N 1 > 0 such that for n > N 1 and for all θ ∈ Θ, u εn (x εn , θ, t εn ) ≤ −δ/4.

4.1.4.
Proof of the auxillary lemma. Now we present the proof of Lemma 4.1.
Proof of Lemma 4.1. We will give the proof of the case that u − v has a local minimum at (x 0 , t 0 ). The proof of the other case is similar. Without loss of generality we assume that u − v has a strict local minimum at (x 0 , t 0 ) in B r (x 0 , t 0 ) for some r > 0. Let (x ε , θ ε , t ε ) be a local minimum of u εn − v εn in B r/2 (x 0 , t 0 ) × Θ.
Since (x ε , t ε ) → (x 0 , t 0 ), the definition of u(x 0 , t 0 ) implies that equality holds in the above. This completes the proof of the lemma.

4.2.
Proof of Part (II) of Proposition 1.2. In the previous subsection we showed thatū and u are a supersolution and subsolution, respectively, of (HJ) on R × (0, ∞). In this section we study the behavior of u and u at time t = 0.
In particular, we have where the equalities follow since v δ (x 0 , 0) = 0 and from (4.18). We now use thatū ≤ 0 and the definition of v δ to estimate the left-hand side of the previous line from above by − |x δ −x0| 2 δ . Thus we obtain an estimate on the distance between x δ and x 0 : (4.20) |x δ − x 0 | 2 δ ≤ M for all δ.
We point out that u(x 0 , 0) is finite. Indeed, if u(x 0 , 0) = −∞ then u − v has a minimum at (x 0 , 0) for all v, and since we know u is a supersolution of (4.24), we find v t + v x c(v x ) ≥ 0 at (x 0 , 0) for all v, which is of course impossible.
Since u is lower-semicontinuous and finite at (x 0 , 0), there exists a neighborhood Q of (x 0 , 0) and some finite M > 0 such that if (x, t) ∈ Q, then u(x, t) > −M .
We define, for δ > 0, the test functions where we define ν by (4.19) as in the previous proof. There exists a sequence (x δ , t δ ) → (x 0 , 0) such that u − v δ has a local minimum at (x δ , t δ ) ∈ Q. We find, as in the previous proof, which implies that the upper bound (4.20) on |x δ − x 0 | 2 holds here as well. We use (4.20), the definition of ν(δ) and the properties of c given in Proposition 1.3 to obtain, Let us suppose t δ > 0. Since u − v δ has a minimum at (x δ , t δ ) and, according to Part (I) of Proposition 1.2, u is a supersolution of (HJ) in R × (0, ∞), we find which is impossible, as we have established (4.28). Therefore, we find t δ = 0 for all δ. According to (4.24), we have that u is a supersolution of (4.24) on K × {0}, so we find According to (4.28) we see that the second term in the max is strictly negative. Hence we obtain u(x δ , 0) ≥ 0 for all δ. Together with v δ (x 0 , 0) = 0, the fact that (x δ , 0) is a minimum of u − v δ , and v δ ≤ 0, this implies, But this contradicts our assumption (4.27). Thus we find u(x 0 , t) ≥ 0 and the proof is complete.

5.
Limits of the n ε as ε approaches zero In this short section we give use the results that we've established in the rest of the paper to give the proof of our main result, Theorem 1. Proof of Theorem 1.2. Since (HJ) satisfies the comparison principle (see Proposition A.1), Proposition 1.2 and the definitions of u J and u K implyū ≤ u J and u ≥ u K on R × (0, ∞).
Recalling the definition of lim sup * completes the proof.
Appendix A.
In this appendix, we state a comparison principle for (HJ) with infinite initial data (Proposition A.1). This result was established in [5, Section 4] but not explicitely stated there, and we could not locate another reference in the literature. Because of this, we carefully explain how to obtain it from [5,Section 4]. Similarly, because we could not locate in the literature the proof of the exact existence result we need, Lemma 1.1, we sketch its proof here.
In addition to [20,5], we also refer the reader to Crandall, Lions and Souganidis [16] for more about Hamilton-Jacobi equations with infinite initial data.
Proposition A.1. Suppose w is a viscosity solution of (HJ) on R × (0, ∞) with with initial data (1.1). Suppose w andw are, respectively, a viscosity subsolution and a viscosity supersolution of (HJ) on R×(0, ∞), and both with the initial data (1.1). Then we have (A.1) w ≤ w ≤w.
Proof. We explain why Proposition A.1 is exactly what was proven in [5,Section 4]. First, we point out that the Hamiltonian H in [5] is assumed to be convex, but this is not used in the arguments of [5,Section 4]. Second, there is a difference in sign between our paper and [5, Section 4], which we now address. We have that −w, −w and −w are, respectively, a viscosity solution, supersolution, and subsolution of min{u, u t + c(−u x )u x } = 0 with initial data 0 for x ∈ Ω ∞ for x ∈ R \Ω.
Thus we see that, if we take H(p) = c(−p)p and G 0 = Ω, then we are exactly in the situation of [5,Section 4]. We may take I = −w, v * = −w, and v * = −w. The conclusion of the comparison of the arguments in [5, Section 4] (specifically, lines (4.2) and (4.5)) is v * ≤ I ≤ v * . Translating back to our notation, we see (A.1) holds.
Proposition A.1 implies that there exists at most one viscosity solution to (HJ) with infinite initial data (1.1). We briefly explain why there exists a solution to (HJ) with infinite initial data (1.1), as we could not find this exact result in the literature (in [20,5] the convex structure of the Hamiltonian is used to establish existence). Hence we provide: