Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.

Equation (1.1) is used to model various diffusive processes in biology, ecology, combustion theory and so on. However, when processes with jumps come into play, equation (1.1) is no longer suitable. More precisely, the random dispersal operator ∂ xx is no long suitable. As a substitute, the nonlocal dispersal operator is introduced (see e.g. [17] for some background) and we are now concerned with the following integral equation where J is a convolution kernel and [J * u](x) = R J(x − y)u(y)dy = R J(y)u(x − y)dy. As for (1.2) in the homogeneous media, traveling waves, i.e., solutions of the form u(t, x) = φ(x − ct) with (c, φ) satisfying have been obtained for various nonlinearities, including bistable nonlinearity, monostable nonlinearity, and ignition type nonlinearity (see [4,9,10,11,12,13,38] and references therein). The study of (1.2) in the heterogeneous media is rather recent and results concerning front propagation are very limited comparing to that of the classical random dispersal case. In [14,47,48,49], the authors investigated (1.2) in the space periodic monostable media, i.e., f (t, x, u) = f (x, u) is of monostable type and periodic in x, and proved the existence of spreading speed and periodic traveling waves. In [37], the authors studied the existence of spreading speeds and traveling waves of (1.2) in the space-time periodic monostable media. Very recently, both Berestycki, Coville and Vo (see [5]), and Lim and Zlatoš (see [26]) investigated (1.2) in the space heterogeneous monostable media. While Berestycki, Coville and Vo stuided principal eigenvalue, positive solution and long-time behavior of solutions, Lim and Zlatoš proved the existence of transition fronts in the sense of Berestycki-Hamel (see [7,8]), that is, an entire solution u(t, x) of (1.1) or (1.2) is called a transition front if u(t, −∞) = 1 and u(t, ∞) = 0 for any t ∈ R, and for any ∈ (0, 1) there holds sup t∈R diam{x ∈ R| ≤ u(t, x) ≤ 1 − } < ∞.
The main result in the present paper is stated in the following theorem.
Moreover, there exists a continuous differentiable function X : R → R such that the following hold: (i) there exist c min > 0 and c max > 0 such that c min ≤Ẋ(t) ≤ c max for all t ∈ R; (ii) there exist two exponents c ± > 0 and two shifts h ± > 0 such that Clearly, due to space homogeneity, any space translation of u(t, x) is also a transition front. Considering space reflection, we obtain space increasing transition fronts. Note that if u(t, x) is continuously differentiable, then φ(t, x) := u(t, x + X(t)) satisfies In the literature, u(t, x) = φ(t, x − X(t)) is also called a generalized traveling wave, especially in the time heterogeneous media (see e.g. [32,39,40,43] (i) We first construct space decreasing front-like solutions u(t, x; s) of (1.3) with initial moment at s < 0 satisfying the normalization u(0, 0; s) (see Lemmas 2.2 and 2.3). We are going to show that u(t, x; s) converge to a transition front as s → −∞. Unlike in the classical random dispersal case, right now, we cannot conclude the convergence of u(t, x; s) to some entire solution of (1.3) due to the lack of space regularity. In fact, from continuity and monotonicity, we only know that they are a.e. differentiable in space. Thus, in order for the convergence, certain space regularity needs to be established. Also, in order for u(t, x; s) to converge to a transition front, the uniform boundedness of the interface width of u(t, x; s) needs to be established. (ii) For λ ∈ (0, 1), define the interface location X λ (t; s) via u(t, X λ (t; s); s) = λ. One important step in constructing transition fronts is to establish the boundedness of interface width, that is, sup s<0,t≥s for 0 < λ 1 < λ 2 < 1. Combining the rightward propagation estimate of X λ (t; s) and an idea of Zlatoš (see [54,Lemma 2.5]), we are able to show (1.6) for 0 < λ 1 < λ 2 ≤ λ * , where λ * ∈ (θ, 1) is fixed (see Theorem 3.1). Of course, this is not enough to describe the front-like shape of u(t, x; s). We need (1.6) to be true for all 0 < λ 1 < λ 2 < 1. (iii) Another important step in constructing transition fronts is to establish the steepness estimate, that is, u x (t, X θ (t; s); s) is uniformly negative, since then the formulȧ ensures the finite speed of X θ (t; s). However, this does not work here for that u x (t, X θ (t; s); s) may not exist due to the lack of space regularity. To circumvent this, we look at the problem from a different viewpoint. Instead of trying to obtain some properties of X θ (t; s), we modify it to get what we want. As a result, we obtain a new interface location X(t; s), which is continuously differentiable, has bounded and uniform positive derivative and stay within a neighborhood X θ (t; s), and hence, captures the propagation nature of u(t, x; s) (see Theorem 4.1). We then use this new interface location to derive the uniform exponential decaying estimate of u(t, x; s) at ∞ and of 1 − u(t, x; s) at −∞ (see Theorem 4.2). From which, we establish (1.6) for all 0 < λ 1 < λ 2 < 1 (see Corollary 4.3). (iv) We have obtained all the expected properties of u(t, x; s) except one: certain space regularity. Here, we establish the uniform Lipschitz continuity in space, that is, (see Lemma 5.1). The idea of proof is based on the decomposition of R into moving intervals according to the new interface location in (iii). Then, for any fixed x, it can stay in the interval causing the growth of u(t,x+η;s)−u(t,x;s) η only for a certain period of time. For the analysis, we need (1.6). With this regularity and the above mentioned properties, we finally construct a transition front, and finish the proof of Theorem 1.1. Although the interface locations X(t) in Theorem 1.1 have very nice properties, in general, we have no idea about the value u(t, X(t)). To have a better understanding of how u(t, x) propagates, let us look at the interface locations at the ignition temperature, that is, X θ (t) such that u(t, X θ (t)) = θ for all t ∈ R. Due to the space monotonicity of u(t, x), X θ (t) is well-defined and continuous. We prove In particular, lim sup t→±∞ It is known that X θ (t) oscillates due to the time dependence of f . Theorem 1.2 then says that such oscillation is locally uniformly bounded.
The paper is organized as follows. In Section 2, we construct approximating solutions and study some fundamental properties. In Section 3, we show that interface width of approximating solutions remains uniformly bounded as time elapses. In Section 4, we construct modified interface location and use it to derive uniform exponential decaying estimates of approximating solutions. In the final section, Section 5, we study the uniform Lipschitz continuity of approximating solutions and finish the proof of Theorem 1.1 and Theorem 1.2. The paper is ended up with Appendix A on comparison principles.

Approximating front-like solutions
In this section, we construct approximating front-like solutions of (1.3), which will be shown to converge to a transition solution of (1.3). Throughout this section, we assume (H1) and (H2). For the comparison principle, it is referred to Proposition A.1 in the appendix.
First we note that, by general semigroup theory (see e.g. [36]), for any u 0 ∈ C b unif (R, R) and s ∈ R, (1.3) has a unique (local) solution u(t, ·; s, u 0 ) ∈ C b unif (R, R) with u(s, x; s, u 0 ) = u 0 (x), where C b unif (R, R) = {u ∈ C(R, R) | u is uniformly continuous on R and sup x∈R |u(x)| < ∞} equipped with the norm u = sup x∈R |u(x)|. Moreover, u(t, ·; s, u 0 ) is continuous in s ∈ R and u 0 ∈ C b unif (R, R). By the comparison principle, if u 0 (x) ≥ 0 for x ∈ R, then u(t, ·; s, u 0 ) exists for all t ≥ s and u(t, x; s, u 0 ) ≥ 0 for t ≥ s and x ∈ R.
Consider the following homogeneous equation where f min , given in (H2), is of ignition type. Let us first summarize some results obtained in [11]. There are a unique c min > 0 and a unique continuously differentiable function φ = φ min : R → (0, 1) satisfying That is, φ min is the normalized wave profile and φ min (x − c min t) is the traveling wave of (2.1). Moreover, φ min ∈ C 1 (R) and it enjoys exponential decaying estimates stated in the following lemma.
We remark that it is shown in [11] that φ min does not decay faster than exponential function at ∞ and 1 − φ min does not decay faster than exponential function at −∞.
For s < 0 and y ∈ R, denote by u(t, x; s, φ min (· − y)) the classical solution of (1.3) with initial data u(s, x; s, φ min (· − y)) = φ min (x − y). We remark that there is no guarantee of space regularity of u(t, x; s, φ min (· − y)). The next lemma gives the approximating solutions.
Moreover, the above analysis implies that and hence, y s → −∞ as s → −∞.
For notational simplicity, in what follows, we will always write u(t, x; s, φ min (· − y s )) as u(t, x; s). Thus, u(s, ·; s) = φ min (· − y s ). The next lemma provides some fundamental results of u(t, x; s). (ii) u(t, x; s) is strictly decreasing in x. In particular, u(t, x; s) is almost everywhere differentiable in x.
Proof. (i) It follows from the fact u(t, x; s) ∈ (0, 1) by the comparison principle, the estimate (2.3) and the following estimate for some sufficiently large c > 0, which is derived in Lemma 2.2.
(ii) For the monotonicity, we first see that u(s, x; s) = φ min (x − y s ) is strictly decreasing in x. For any y > 0, we apply comparison principle to u(t, x − y; s) − u(t, x; s) to conclude that that u(t, x − y; s) > u(t, x; s) for t > s. The result then follows.
We remark that the family {u(t, x; s)} s<0 is the approximating front-like solutions, which will be shown to converge to a transition front of (1.3) as s → −∞. However, due to the lack of space regularity as mentioned before, it is not clear that this family will converge to some solution of (1.3). Also even if u(t, x; s) converges to some solution of (1.3), it is difficult to see that the limiting solution is a transition front. We will then first establish in Sections 3 and 4 the uniform boundedness of the interface width of u(t, x; s) and uniform decaying estimates of u(t, x; s), respectively, which assure the limiting solution of u(t, x; s) as s → −∞ (if exists) is a transition front. Later, in Lemma 5.1, we will show the uniform Lipschitz continuity in space of the approximating solutions, which of course implies the convergence of the approximating solutions thanks to Arzelà-Ascoli theorem, but its proof, using the uniform boundedness of interface width (see Corollary 4.3), is not straightforward.

Bounded interface width
For s < 0, t ≥ s and λ ∈ (0, 1), let X λ (t; s) be such that u(t, X λ (t; s); s) = λ. By Lemma 2.3, it is well-defined and continuous in t. Moreover, X λ 1 (t; s) > X λ 2 (t; s) if λ 1 < λ 2 . We point out that X λ (t; s) is not monotonic in t due to the time dependence of f . As usual, we refer to X λ (t; s) as the interface location, and the corresponding point on the solution curve as the interface. Assume (H1) and (H2) in this section. The main result in this section is stated in the following theorem.
Theorem 3.1. There exists λ * ∈ (θ, 1) such that for any 0 < λ 1 < λ 2 ≤ λ * , there holds sup s<0,t≥s Theorem 3.1 shows the boundedness of the width between interfaces below λ * . Later in Corollary 4.3, it is extended to any 0 < λ 1 < λ 2 < 1. The proof of Theorem 3.1 is long, and therefore, broken into several parts. In Subsection 3.1, we give rightward propagation estimates of interfaces above the ignition temperature. In Subsection 3.2, we define new interface locations Y κ (t; s) and establish the uniform boundedness between Y κ (t; s) and X λ (t; s). We finish the proof of Theorem 3.1 in Subsection 3.3.
3.1. Rightward propagation estimates of interface locations. In this subsection, we study the rightward propagation nature of X λ (t; s). To do so, we need some knowledge on the traveling waves of It is proven in [4] that the problem admits a unique solution (c B , φ B ) with c B > 0. Moreover, the following stability result holds: Then, there exists The main result in this subsection is stated in the following proposition.
Since f (t, u) ≥ f min (u) and u(t 0 , · + X λ (t 0 ; s); s) ≥ u 0 , the comparison principle implies that We now show (3.5). Since u 0 (x) = λ for x ≤ x 0 , continuity with respect to the initial data (in sup norm) implies that for any > 0 there exists δ > 0 such that where the equality is due to monotonicity. By (H1), J concentrates near 0 and decays very fast as x → ±∞. Thus, we can choose if we choose > 0 sufficiently small, since then f min (u min (t, x; u 0 )) is close to f min (λ), which is positive. This simply means that u min (t, x; u 0 ) > λ for all x ≤ x 1 and t ∈ (0, δ], which implies that ξ min (t) > x 1 for t ∈ (0, δ]. The continuity of ξ min then leads to (3.5). This proves (3.4). The result of the proposition then follows from (3.3) and (3.4).

3.2.
Auxiliary interface width estimates. In this subsection, we define a new interface location and study the boundedness between this new interface location and the original ones. For κ > 0, set c κ := inf λ>0 1 λ R J(y)e λy dy − 1 + κ > 0. It is not hard to see that there exists a unique λ κ > 0 such that We point out that c κ is the minimal speed of the traveling waves of the following KPP equation, with f KPP (0) = κ, where f KPP (0) = 0 and f KPP (u)/u is strictly decreasing in u ≥ 0. But a proof is needed in the case that J is not compactly supported (see e.g. [9,12]). In fact, in the case that J is not compactly supported, it is known that c KPP ≤ κ (see [12]). As in the classical random dispersal case, we have Lemma 3.4. c κ → 0 and λ κ → 0 as κ → 0.
Proof. We see We show λ κ → 0 as κ → 0. It is understood that λ κ is the unique positive point such that This simply means that the unique solution of g(λ) = κ goes to 0 as κ → 0. This completes the proof.
By the definition ofc κ , we readily check that v t = J * v − v + κ 0 v. By the definition of κ 0 , we have κ 0 v ≥ f sup (v) for all v ≥ 0. It then follows from v(t 0 , x; t 0 ) = e −λκ(x−Yκ(t 0 ;s)) ≥ u(t 0 , x; s) by (3.7) and the comparison principle that v(t, x; t 0 ) ≥ u(t, x; s) for t ≥ t 0 , which leads to the result.
We now prove the main result in this subsection. In what follows, we fix some small κ > 0 such that c κ < c B , and for this fixed κ, we write Y κ (t; s) as Y (t; s), and set λ * := min u > 0 f max (u) = κu ∈ (θ, 1). (3.8) for all s < 0, t ≥ s.
Proof. From the definition of λ * , we readily see that Fix an λ ∈ (θ, λ * ]. Set C 0 = max{Y (s; s) − X λ (s; s), 1}. We see that C 0 is independent of s < 0. This is because that u(s, ·; s) = φ min (· − y s ), and hence, space translations do not change Y (s; s) − X λ (s; s). Clearly, we have the estimate sup s<0,t≥s [X λ (t; s) − Y (t; s)] ≤ C for some large C > 0. Set = c B −cκ 2 and C 1 = C 0 + c B t ,λ , where t ,λ is as in Proposition 3.3. To finish the proof, we only need to show sup s<0,t≥s [Y (t; s)−X λ (t; s)] ≤ C 1 . Suppose this is not the case, then we can find some We claim Y (t 0 ; s 1 ) − X λ (t 0 ; s 1 ) ≤ C 0 . It is trivial if there are only finitely many t ∈ [s 1 , t 1 ] such that Y (t; s 1 ) − X λ (t; s 1 ) ≤ C 0 . So we assume there are infinitely many such t and the claim is false. Then, there exists a sequence {t n } n∈N ⊂ [s 1 , t 0 ) such that Y (t n ; s 1 ) − X λ (t n ; s 1 ) ≤ C 0 for n ∈ N andt n → t 0 as n → ∞. Moreover, Y (t 0 ; s 1 )−X λ (t 0 ; s 1 ) =C 1 > C 0 . It then follows that for all n ∈ N where the second inequality is due to Lemma 3.5. Passing n → ∞, we easily deduce a contradiction from the continuity of X λ (t; s 1 ). Hence, the claim is true, that is, We show Y (t 0 ; s 1 ) − X λ (t 0 ; s 1 ) = C 0 .
Note, by Lemma 2.3 and the definition of v(t, x; t 0 ), the limit v(t, x; t 0 ) − u(t, x; s 1 ) → 0 as x → ∞ is uniformly in t ∈ [t 0 , t 2 ]. Then, applying the comparison principle (see Proposition A.1) to v(t, x; t 0 ) − u(t, x; s 1 ), we conclude It follows that Y (t; s 1 ) ≤Ỹ (t; s 1 ) for t ∈ [t 0 , t 2 ] by definition in (3.7). In particular, It is a contradiction. Thus, the claim follows, that is, X λ (t; s 1 ) <Ỹ (t; s 1 ) for all t ∈ [t 0 , t 1 ], and repeating the above arguments, we see It follows from (3.11) and Proposition 3.3 that for any t ∈ [t 0 , t 1 ] This is a contradiction. Consequently, Y (t; s) − X λ (t; s) ≤ C 1 for all s < 0, t ≥ s. This completes the proof.
We remark that Proposition 3.6 is a nonlocal version of [45,Lemma 3.4], whose proof is based on the rightward propagation estimate as in Proposition 3.3 and an idea of Zlatoš (see [54,Lemma 2.5]).

Modified interface locations and decaying estimates
In the study of the propagation of the solution u(t, x; s), the propagation of the interface location X λ (t; s), more precisely, how fast it moves, plays a crucial role. In the classical random dispersal case, this problem is transferred into the study of uniform steepness, that is, whether u x (t, X λ (t; s); s) is uniformly negative, since there holds the formulȧ Clearly, this approach does not work here since we are lack of space regularity of u(t, x; s). Moreover, we do not know if X λ (t; s) is differentiable in t and it moves back and forth in general. To circumvent these difficulties, we look at the problem from a different viewpoint. Instead of studying X λ (t; s) directly, we modify it to get a new interface location of expected properties, such as moving in one direction with certain speed and staying within a certain distance from X λ (t; s), which capture the propagation nature of u(t, x; s). This is the purpose of Subsection 4.1. As an application of the new interface location, we derive uniform exponential decaying estimates of u(t, x; s) in Subsection 4.2.

Modified interface locations.
In this subsection, we modify X λ (t; s) properly and prove the following theorem.
Proof. By Proposition 3.3, there exists t B > 0 such that Recall Y (t; s) is Y κ (t; s) for some fixed small κ > 0 and we have by Proposition 3.6, and by Lemma 3.5, where c 0 =c κ for the fixed κ > 0. We interpret that (4.1), (4.2) and (4.3) imply that X λ * (t; s) moves with a uniformly positive and uniformly bounded average speed. This observation is crucial in the following modification.

4.2.
Uniform exponential decaying estimates. In this subsection, we study uniform exponential decaying estimates of u(t, x; s) behind and ahead of interfaces. Let λ * be as in (3.8) and X(t; s) be as in Theorem 4.1. We see that there exist θ * ∈ (θ, λ * ] and β > 0 such that We remark that β may be very small. Set whereĈ > 0 is some constant (to be chosen) introduced only for certain flexibility. Theorem 3.1 and Theorem 4.1 then imply thatX(t; s) ≤ X θ * (t; s), and hence, u(t, x +X(t; s); s) ≥ θ * for all x ≤ 0. We also set Since X(t; s) ≥ X λ * (t; s) by Theorem 4.1, we haveX(t; s) ≥ X θ (t; s), and hence, u(t, x + X(t; s); s) ≤ θ for x ≥ 0. We now prove the main result in this subsection.
There exist c ± > 0 such that For x ≤X(t; s), we have u(t, x; s) ≥ θ * , which together with (4.5) implies that It then follows that for x ≤X(t; s) For c > 0, we compute where we used the definition ofX(t; s) and Theorem 4.1. Since R J(y)e −cy dy → 1 as c → 0, we can choose c > 0 so small that cc max + R J(y)e −cy dy − 1 − β ≤ 0, and then, N − [1 − e c(x−X(t;s)) ] ≤ 0. Hence, we have shown for some small c − > 0. Trivially, we have u(t, x; s) > 0 ≥ 1 − e c − (x−X(t;s)) for x ≥X(t; s). At the initial moment s, we obtain from Lemma 2.1 that u(s, x; s) = φ min (x − y s ) ≥ 1 − e c − (x−X(s;s)) if we choose c − smaller andĈ sufficiently large. Then, we conclude from the comparison principle (see Proposition A.1(ii)) that u(t, x; s) ≤ 1−e c − (x−X(t;s)) for x ≤X(t; s). This proves half of the theorem.
We remark that the fact thatẊ(t; s) ≥ c B 2 plays an essential role in deriving the uniform exponential decaying estimate of u(t, x; s) at ∞.
The result then follows from the fact thatX(t; s) −X(t; s) ≡ const.

Construction of transition fronts
In this section, we prove Theorem 1.1 and Theorem 1.2. To do so, we prove uniform Lipschitz continuity in space of u(t, x; s).  Thus, to finish the proof of the lemma, it suffices to show the locally uniform Lipschitz continuity, that is, To this end, we fix δ > 0. Let X(t; s) be as in Theorem 4.1 and define Xθ(t; s) − X(t; s) , whereθ ∈ (θ, 1) is given in (H2). Notice L 1 < ∞ and L 2 < ∞ by Corollary 4.3. Then, for any 0 < |y − x| ≤ δ we have • if x ≥ X(t; s) + L 1 , then y ≥ x − δ ≥ X θ (t; s), which implies that f (t, u(t, y; s)) = 0 = f (t, u(t, x; s)), and hence f (t, u(t, y; s)) − f (t, u(t, x; s)) u(t, y; s) − u(t, x; s) = 0; (5.2) • if x ≤ X(t; s) − L 2 , then y ≤ x + δ ≤ Xθ(t; s), which implies that u(t, y; s) ≥θ and u(t, x; s) ≥θ, and hence by (H2) f (t, u(t, y; s)) − f (t, u(t, x; s)) u(t, y; s) − u(t, x; s) ≤ 0. Since X(t; s) is continuous in t, this three regions change continuously in t. As X(t; s) moves to the right by Theorem 4.1, any fixed point will eventually enter into R l (t; s) and stay there forever. For s < 0 and x 0 ∈ R, let t first (x 0 ; s) be the first time that x 0 is in R m (t; s), that is, and t last (x 0 ; s) be the last time that x 0 is in R m (t; s), that is, Since X(t; s) moves to the right, if x 0 ∈ R l (s; s), then x 0 ∈ R l (t; s) for all t > s. In this case, t first (x 0 ; s) and t last (x 0 ; s) are not well-defined, but it will not cause any trouble. Clearly, x 0 ∈ R l (t; s) for all t > t last (x 0 ; s). We see that either both t first (x 0 ; s) and t last (x 0 ; s) are well-defined, or both of them are not well-defined. In fact, t first (x 0 ; s) and t last (x 0 ; s) are well-defined only if x 0 / ∈ R l (s; s). As a simple consequence of Proposition 3.3 and the fact that the length of R m (t; s) is L 1 + L 2 , we have T = T (δ) := sup Now, we are ready to prove the lemma. Fix x 0 ∈ R, s < 0 and 0 < |η| ≤ δ. Set v y (t, x; s) = u(t, x + η; s) − u(t, x; s) η .

Appendix A. Comparison principles
We prove comparison principles used in the previous sections.
Proposition A.1. Let K : R × R → [0, ∞) be continuous and satisfy sup x∈R R K(x, y)dy < ∞. Let a : R × R → R be continuous and uniformly bounded.
Proof. (i) We follow [20,Proposition 2.4]. Note first that replacing u(t, x) by e rt u(t, x) for sufficiently large r > 0, we many assume, without loss of generality, that a(t, x) > 0 for all (t, x) ∈ R × R.