Asymptotic Behavior of a Nonlocal KPP Equation with an Almost Periodic Nonlinearity

We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.


Introduction
The aim of this paper is to analyze the large space/long time asymptotic behavior of the nonlocal reaction-diffusion equation where J is a continuous, nonnegative, compactly supported, and symmetric kernel, and f is a monostable (KPP) type nonlinearity in u for which the canonical example is f (u) = u(1 − u). To study the asymptotic behavior of (1.1), we introduce the "hyperbolic" scaling (x, t) → (ǫ −1 x, ǫ −1 t).
As ǫ → 0, the time scaling reproduces long-time behavior of (1.1), while the space scaling reproduces in bounded sets behavior for large space variables. The new unknown is now given by u ǫ (x, t) := u(ǫ −1 x, ǫ −1 t). We introduce an initial condition u ǫ (·, 0) = u 0 (·), and we can easily see that u ǫ satisfies The behavior of u ǫ as ǫ → 0 is what we will consider to determine the asymptotic behavior of (1.1).
To obtain a result concerning the convergence of u ǫ , it is necessary to make assumptions about the oscillatory behavior of f , and in this paper we assume that f is an almost-periodic function in the x ǫ variable. Our main result, Theorem 3.1, states that as ǫ → 0, u ǫ respectively converges to the two equilibria of f , which for simplicity we take to be constant, in the two regions {φ < 0} and int({φ = 0}), where φ is the solution of the Hamilton-Jacobi variational inequality G 0 is the support of u 0 , and H(p) is an "effective Hamiltonian" resulting from the homogenization of (1.2). This behavior was shown for a nonlocal equation very close to (1.1) that models the propagation of an invasive species in ecology by Perthame and Souganidis in [24], and similar asymptotic behavior was found for a non-local Lotka-Volterra equation by Barles, Mirrahimi, and Perthame in [8].
Because the behavior of the solutions of (1.1) consists of two equilibrium states joined together by a transition layer near the interface defined by (1.3), and the effective Hamiltonian H(p) can be interpreted as the propagation speed of this interface, our work is connected with the wellstudied areas of traveling wave solutions of the KPP equation and the speed of their associated traveling fronts. Recent articles concerning these aspects of nonlocal KPP equations include those by Coville, Dávila, and Martínez, who in [10] and [11] studied (1.1) in the case where f is periodic in x. They showed that there exists a critical speed which is the lowest speed for which there exists a pulsating front solution of (1.1). The existence of traveling wave solutions and of a critical speed was considered for a non-local KPP equation similar to (1.1) by Berestycki, Nadin, Perthame, and Ryzhik in [9]. Lim and Zlatos in [21] gave conditions on the inhomogeneity of f in order to prove existence or non-existence of transition fronts for (1.1), where they also studied the range of speeds for which transition fronts exist.
The local version of (1.1), i.e. the equation where the integral term is replaced by a uniformly elliptic second-order operator, has been studied extensively. Its rescaled form reads It was originally studied in the 1930's by Fisher in [16] and by Kolmogorov, Petrovskii, and Piskunov in [20]. Freidlin in [17] studied the behavior of (1.4) using probabilistic methods for the x ǫ -independent problem. Evans and Souganidis in [14] extended [17] and introduced a different approach based on PDE methods which has proven to be more flexible. The asymptotic behavior of u ǫ in the presence of periodic space-time oscillation was analyzed by Majda and Souganidis in [23]. Our work is an extension of [14] and [23] to the nonlocal case. There is also a vast literature dealing with the long-time behavior of (1.4), going back to the work of Aronson and Weinberger [5].
Due to the presence of the oscillatory variable x ǫ in (1.2), the theory of homogenization plays a crucial part in the analysis of this equation as ǫ → 0. The study of homogenization of Hamilton-Jacobi equations in periodic settings began with the work of Lions, Papanicolaou, and Varadhan [22], and homogenization for "viscous" Hamilton-Jacobi equations was studied by Evans [15] and Majda and Souganidis [23]. The fundamental tool in the periodic setting is the fact that it is possible to solve the macroscopic problem, or "cell problem." Homogenization in the almost-periodic case was established by Ishii [18], who used the almost periodic structure to construct approximate correctors.
Arisawa in [3] and [4] studied the periodic homogenization of integro-differential equations with Lévy operators, equations that are similar in structure to the ones we consider, and we employ the general ideas of her work. She considered the "ergodic problem", which is the same as the cell problem, and proved that approximate correctors exist by considering the limit along a subsequence of a family of functions that satisfy an approximated cell problem and showing that the limiting equation satisfies a strong maximum principle. Since such a limiting equation and strong maximum principle are not available in our case, we will use more direct techniques based on an analysis of the nonlocal term to prove the existence of the approximate corrector, and we show that almostperiodicity provides enough of a "compactness" criterion in order to make these techniques work.
The paper is organized as follows. In Section 2, we make precise our assumptions. In Section 3, we state our main result, Theorem 3.1, and we give a heuristic justification for it. In Section 4, we give the proof of homogenization, and in Section 5, we finish the proof of Theorem 3.1 using the homogenization result.

Assumptions
We assume that f : R n+1 → R is smooth and has bounded derivatives. In particular, it satisfies x,u f (x, u)|} < ∞ for each L > 0, D x,u denotes derivatives with respect to x and u; in the rest of this paper D denotes derivatives with respect to the space variable x. We also assume that f is of KPP type i.e. monostable in the u variable. That is, for every x ∈ R n , f satisfies Because (2.1) implies that f (x, u) is smooth and has locally in u and globally in x bounded first and second derivatives in both variables, we can see that c(x) is smooth, bounded, and Lipschitz continuous.
Concerning the kernel J, we assume that Concerning the initial condition u 0 , we assume that We assume that the nonlinearity is almost-periodic, that is, we assume that the family Note that the typical assumption of 1-periodicity is a specific case of almost-periodicity.

Main Result, Heuristic Derivation
We now state our main result. Next we explain in a heuristic way the origin of the variational inequality and why it controls the asymptotic behavior of the u ǫ . Following the work for local KPP equations mentioned in the introduction, we now use the classical Hopf-Cole transformation It is immediate that for t = 0, φ ǫ = −∞ on R n \G 0 and φ ǫ → 0 on G 0 as ǫ → 0. The interesting part of the transformation comes into play for t > 0. We can see via straightforward calculations that φ ǫ solves an equation which can be analyzed using homogenization techniques. We assume that φ ǫ admits the asymptotic expansion φ ǫ (x, t) = φ(x, t) + ǫv( x ǫ ) + O(ǫ 2 ). Writing z = x ǫ and performing a formal computation, we obtain Formally, we can say that as ǫ → 0, ǫ −1 (φ(x − ǫy, t) − φ(x, t)) → −y · Dφ(x, t). In addition, if u ǫ → 0 as ǫ → 0, then Writing p = Dφ(x, t), we see that oscillatory behavior disappears in the limit as ǫ → 0 if it is possible to find a constant H(p) and a function v that solves which is a typical macroscopic problem or "cell problem" from homogenization theory. The issue is to find H(p), referred to as the effective Hamiltonian, so that (3.3) admits a solution v, typically referred to as a "corrector," with appropriate behavior at infinity i.e. strict sublinearity, so that H(p) is unique. If an effective Hamiltonian and a corresponding corrector can be found, then we see that φ ǫ converges to a function φ that satisfies φ t + H(Dφ) = 0, provided that we also ensure that φ < 0 so u ǫ → 0 due to (3.1), which would then allow us to apply (3.2). Therefore, φ should satisfy the Hamilton-Jacobi variational inequality which combined with the initial condition at t = 0 is precisely (1.3). Then (3.1), the fact that φ ǫ → φ, and an additional argument, found in Section 5, to show that u ǫ → 1 on the set {φ = 0} imply that u ǫ satisfies the behavior described by Theorem 3.1.

Proof of Homogenization
We proceed to prove Theorem 3.1 rigorously. Our primary result in this section is the homogenization of (4.1), that is, we show that solutions φ ǫ of (4.1)  Our first objective is to find H(p) such that the cell problem (3.3) admits "approximate correctors" v + , v − that satisfy (4.6) and (4.7) respectively, as the existence of approximate correctors is sufficient to prove homogenization. The proofs in the almost-periodic and periodic cases are very similar, so we will present the proof in the almost periodic case and explain how the proof differs in the periodic case.
We start by making the typical approximation to the cell problem (see [22]) and consider the following equation in R n for λ > 0: First we need to show that this problem is well-posed. The proof follows along similar lines of other comparison proofs (see [1], [2], [6], [7], [12], [19]).
Proposition 4.2. Let u(z) ∈ USC(R n ) be a bounded subsolution of (4.2), and let v(z) ∈ LSC(R n ) be a bounded supersolution of (4.2). Then u ≤ v in R n . In addition, there exists a unique bounded continuous solution of (4.2).
Proof. We first prove that comparison holds. Assume for a contradiction that M := sup z∈R n u(z) − v(z) > 0. Then define Note that this quantity is positive for δ sufficiently small. Because of our assumption that u, v are bounded, there exists a point z δ such that Because u, v are respectively a subsolution and a supersolution of (4.2), we obtain Subtracting the second inequality from the first, we have We know that for any y ∈ R n , (4.3) implies that as δ → 0, for any y ∈ R n , δ[2z δ · y − |y| 2 ] → 0. Therefore, because u, v are bounded, and y is contained in a ball B(0,r) in the integral term of (4.4), we can apply (4.5) to (4.4) and take the limit , but this is a contradiction because the left hand side is uniformly positive by (4.3). Therefore, M ≤ 0, which was what we wanted to show, and this completes the proof of comparison/uniqueness. The existence of a bounded continuous solution given a comparison principle is a consequence of Perron's method, and follows in the same way as the analogous result in [1].
The next proposition, which shows that there exist approximate correctors to the cell problem, is the main objective of this section. It is similar to the analogous one found in [3]. In that work Arisawa considers the "ergodic problem", which is essentially the statement of Proposition 4.4, for a different nonlocal equation, a periodic integro-differential equation containing a Lèvy operator, that bears resemblance to (4.2). In our proof, we will employ some techniques from [3] along with some new ones involving an analysis of the nonlocal term of (4.2).
For the almost periodic setting, we introduce the concept of "uniform almost periodicity".
Definition 4.3. The collection of functions {f s } s∈I for I an arbitrary index set is uniformly almost periodic in s if given any sequence {x j } ∈ R n there exists a subsequence, also called {x j } for convenience, such that for any ǫ > 0 there exists N such that for all s ∈ I and all j, k ≥ N , This definition means that for {f s } the almost periodicity condition (2.6) holds uniformly in s. This concept will be used to give sufficient "compactness" for the almost periodic setting in order to apply the techniques that are applicable to the periodic setting.
In addition, Proof. We first show that if there exists a constant H(p) such that for every ν > 0, there exist functions v + , v − satisfying Proposition 4.4, then H(p) is unique. This argument was originally found in [22]. Proof. Suppose for a contradiction that there exists A < B such that for any ν > 0 there exist bounded v + ν , v − ν that satisfy the following for all z ∈ R n : Fix a sufficiently small ν such that In addition, for ǫ sufficiently small, holds for all z ∈ R n . Now comparison, which can be applied for (4.9) because v + ν and v − ν are bounded and Lipschitz continuous, now implies that v − ν ≥ v + ν . This is a contradiction, and thus B = A and so H is unique. Now we proceed with proving that there exists such a constant H(p). Let z 0 ∈ R n be fixed, and define w λ (z) := v λ (z)−v λ (z 0 ) and C λ := λ w λ ∞ . We know that C λ is bounded due to comparison for (4.2) between λw λ and constant functions depending on sup R n |c(·)|. We claim that if C λ → 0 as λ → 0, then the proposition follows. This is true because λv λ (z) − λv λ (z 0 ) ∞ = λ w λ ∞ → 0. Because λv λ (z 0 ) is bounded in λ, there exists a subsequence such that we can define H(p) := lim λ→0 −λv λ (z 0 ), such that (4.8) holds. Now we can see that upon taking λ sufficiently small so that v λ − H(p) ∞ < ν, v λ satisfies (4.6) and (4.7). Then Lemma 4.5 allows us to finish the proof of Proposition 4.4 in this case.
Therefore, suppose for a contradiction that lim inf λ→0 C λ > 0. Since C λ is uniformly bounded, we can extract a sequence λ n → 0 such that lim n→∞ C λn = C ′ > 0. We will subsequently call this subsequence λ for convenience. Now definẽ Then we have that upon writingc(z) = c(z) − λv λ (z 0 ),w λ satisfies (4.10) Our objective is to show thatw λ converges uniformly to zero. Assume for a contradiction that there exist sequences λ j , z j such that λ j → 0 andw λ j (z j ) → δ = 0, and suppose without loss of generality that δ is positive. We claim that there exists a pointẑ such that for all j sufficiently large, In the case where c(z) is periodic,w λ is also periodic, and then a pointẑ satisfying (4.11) can be found by compactness because the sequence {z j } can be taken to lie in the unit cube. In the case where c(z) is almost-periodic, we use the fact that the family {w λ } λ≤1 is uniformly almost periodic in λ in the sense of Definition 4.3. This is true because by separated z dependence and comparison for (4.10), we know that for any x 1 , x 2 ∈ R n and any λ ≤ 1, there exists a uniform constant C such that and now becausec(z) is uniformly almost periodic in λ, which follows from the fact that c(z) is an almost periodic function, we have that the family {w λ } λ≤1 is uniformly almost periodic. Because {w λ } λ≤1 is uniformly almost periodic, we can extract a subsequence of {z j }, also called {z j }, and take N sufficiently large so that (4.12) |w λ (z j + z) −w λ (z k + z)| ≤ δ 2 for all j, k ≥ N , all z ∈ R n , and for all λ ≤ 1. Now if we fix k ≥ N , then for j sufficiently large, the fact thatw λ (z j ) → δ and (4.12) applied with z = 0 impliesw λ j (z k ) ≥ δ 4 , so z k is a pointẑ that satisfies (4.11). Now we use (4.11) andw λ (z 0 ) = 0 to reach a contradiction. Note that since the integrand of the nonlocal term is always nonnegative, we can restrict our integration domain to suitable subsets when seeking lower bounds. We consider (4.10) as λ j → 0. We have that λw λ → 0 uniformly because w λ ∞ = 1, and there exists a constant C < ∞ such that Therefore we consider the nonlocal second term If we consider the line betweenẑ and z 0 and cover it with M := 3|ẑ−z 0 | r 1 balls of radius r 1 3 , then becausew λ j increases by at least δ 4 on that line from z 0 toẑ, then there exists x j ∈ R n such that Write A j = B(x j , r 1 3 ), and consider x j,min , x j,max ∈ A j to be the points respectively wherew λ j is minimized and maximized over A j . Then we know thatw λ j (x j,max )−w λ j (x j,min ) ≥ δ 2 . In addition, we have due to comparison for (4.10) and the separated z dependence,w λ is Lipschitz continuous with constant K 2 = 2K C ′ for all λ sufficiently small. This gives us Finally, to obtain a contradiction, we consider W λ j (z) with z = x j,min , and we define A 2 := B(x j,min − x j,max , min(r 2 , r 1 − |x j,min − x j,max |)). A 2 is contained in B(0, r 1 ) by construction, and its radius is positive because |x j,min − x j,max | < 2r 1 3 . In addition, for y ∈ A 2 , (4.13) implies that w λ j (x j,min − y) −w λ j (x j,min ) ≥ δ 2 2 . This gives us where C 1 , C 2 > 0 are constants. C 1 λ j exp( C 2 λ j ) is unbounded as λ j → 0, which means that W λ is unbounded, a contradiction to (4.10). Therefore,w λ converges uniformly to zero as λ → 0, but w λ ∞ = 1 for all λ by construction, a contradiction. Therefore, C λ → 0, and we have the existence of H satisfying (4.8). In addition, given ν > 0, we also have the existence of bounded, Lipschitz continuous v + and v − satisfying (4.6) and (4.7) respectively, because we can simply take v + = v − = v λ for λ sufficiently small depending on ν. This concludes the proof of Proposition 4.4 because this means that any convergent subsequence (in the uniform metric) of λv λ (·) must converge to −H(p), and so the full sequence λv λ (x) converges uniformly to −H(p), which is unique by Lemma 4.5.
Remark 4.6. Arisawa in [3] proves that the analogue tow λ in her (periodic) setting converges uniformly to zero by using the uniform equicontinuity ofw λ to find a limiting functionw along a subsequence as λ → 0 via Arzela-Ascoli. She subsequently shows thatw λ → 0 by proving that w solves an equation that has a strong maximum principle. The techniques demonstrated in the above proof overcome the fact that in our case taking λ → 0 in (4.10) does not yield such an equation.
We now move to the proof of Theorem 4.1. We first prove a technical lemma which supplies bounds on u ǫ , the solution of (1.2), and φ ǫ , the solution of (4.1). Note that because φ ǫ is given by (3.1), since we know that (1.2) is well posed (see [1]), and in particular that a comparison principle holds, we know that a comparison principle holds for (4.1) as well.
Proof. The first part is a consequence of comparison for (1.2) and the fact that the constant functions 0 and 1 are respectively a subsolution and a supersolution of (1.2). To prove the second part, note that it suffices to show a lower bound because u ǫ ≤ 1 implies that φ ǫ ≤ 0. To do this, we adapt the argument from Lemma 2.1 of [14]. First, we can assume without loss of generality that there exists an R > 0 such that B(0, R) ⊂ int(G 0 ) and inf B(0,R) u 0 > 0. We first show that φ ǫ is bounded from below on B(0, R) × (0, ∞). To this end, define the function ϕ 1 : , where α, β are positive constants to be chosen. We can now compute where the second inequality follows due to (2.3). Therefore, upon taking α sufficiently large, we can make the right hand side negative on B(0, R) × (0, ∞). If we take β = log(inf B(0,R) u 0 ), then we have that ϕ 1 ≤ φ ǫ on B(0, R) c × (0, ∞) ∪ B(0, R) × {0}. Now comparison for (4.1) implies that ϕ 1 ≤ φ ǫ on B(0, R) × (0, ∞), which means that We now provide a lower bound for the points (x, t) ∈ R n × (0, ∞) such that |x| > R 2 . Define where γ, σ, τ are positive constants to be determined. We can compute We justify the last inequality. It suffices to show that By (2.4) there exists r 2 > 0 such that so by the symmetry of J, we know that there is a positive constant C 3 such that Therefore, if we take γ sufficiently large, we have that (4.16) is satisfied, and thus ϕ 2 is a subsolution of (4.1). Select τ to be larger than the constant from (4.15), and defineφ ǫ (x, t) := φ ǫ (x, t + ξ) for ξ > 0. Then we have that This means that we can apply comparison to conclude that ϕ 2 ≤φ ǫ on B(0, R 2 ) c × (0, ∞), and taking ξ → 0 gives us (4.14).
We will now use the perturbed test function method (see [15], [4]) to prove Theorem 4.1.
We now discuss the properties of the effective Hamiltonian H. In the case of a homogeneous nonlinearity f i.e. c(z) ≡ c, constant functions are correctors, so we can write the form of H(p) to be In particular, we can see that in this situation, the effective Hamiltonian is concave, negatively coercive, and continuous in p, and we now prove that these properties of H(p) hold in general. (1) p → H(p) is concave.
(2) There exist positive constants K 1 , K 2 , K 3 , K 4 > 0, C 1 , C 2 such that H(p) satisfies for all p ∈ R n . In particular, this implies that H is uniformly and negatively coercive.
To show (4.28), we prove a lemma giving a modulus of continuity estimate for v λ .

Convergence of u ǫ
We will now prove Theorem 3.1. That is, we show that as ǫ → 0, u ǫ , the solution of (1.2), converges locally uniformly to 0 and 1 in regions determined by φ, the solution of (1.3). This proof is based on [23] and [14].