Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media

The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. It it shown that these transition fronts have exact decaying rates as the space variable tends to infinity. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work (\cite{ShSh14-4}), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.

The interface location function X(t) tells the position of the transition front as time t elapses, while the uniform-in-t limits shows the bounded interface width, that is, Thus, transition fronts are proper generalizations of traveling waves in homogeneous media and periodic (or pulsating) traveling waves in periodic media. Notice if ξ(t) is a bounded function, then X(t)+ξ(t) is also an interface location function. Therefore, interface location functions are not unique. But, it is not hard to check that the difference of two interface location functions is a bounded function. Hence, interface location functions are unique up to addition by bounded functions.
We see that transition fronts can be defined in the same way for more general equations, say, u t = J * u − u + f (t, x, u).
(1.2) Many results have been obtained for (1.2) when f (t, x, u) is of monostable or Fisher-KPP type in various special cases. For example, when f (t, x, u) = f (u), traveling waves and minimal speeds have been studied in [5,9,10,20]. When f (t, x, u) = f (x, u) is periodic in x or f (t, x, u) is periodic in both t and x, spreading properties and periodic traveling waves have been studied in [11,19,27,28,29]. When f (t, x, u) = f (x, u), while principal eigenvalue, positive solution and long-time behavior of solutions was studied in [2], transition fronts were shown to exist in [16].
In the present paper, we study transition fronts of (1. 2) when f (t, x, u) = f (t, u), that is, (1.1). To state our results, we define for κ > 0 We see that if a(t) ≡ a > 0 is a constant function, then c κ (t) is nothing but the front location function of traveling waves with speed R J(y)e κy dy−1+a κ of a homogeneous nonlocal Fisher-KPP equation (see e.g. [5,10,20]). Note that since inf t∈R a(t) > 0, c κ (t) is increasing, and since sup (t,u) This function will serve as the interface location function as in the definition of transition fronts. Our first result concerning the existence, regularity and decaying properties of transition fronts of (1.1) is stated in the following theorem. Theorem 1.1. Assume (H1) and (H2). There exists κ 0 > 0 such that for any κ ∈ (0, κ 0 ], there is a transition front u κ (t, x) of (1.1) with interface location function X κ (t) = c κ (t) and satisfying the following properties u κ (t,x+X κ (t)) = −κ uniformly in t ∈ R. Note that the continuity of a transition front u(t, x) in the space variable x is not assumed in the definition of transition fronts. But the space regularity of transition fronts plays an important role in the study of other important properties such as stability and uniqueness of transition fronts. In the random dispersal case, the space regularity of transition fronts follows from parabolic Schauder estimates, while, thanks to the lack of space regularity for the nonlocal dispersal equations (that is, the semigroup generated by the nonlocal dispersal operator has no regularizing effect), a transition front of a nonlocal dispersal equation may not be regular in space. We refer the reader to [1] for the existence of discontinuous traveling waves of u t = J * u − u + f B (u), where f B is a balanced bistable nonlinearity. In [25], we established some very general results on the space regularity of transition fronts of nonlocal dispersal equations. Among others, we proved in [25] the following proposition. 25]). Assume (H1) and (H2). Let w(t, x) be an arbitrary transition front of (1.1) satisfying for some C > 0 and r > 0. Then, w(t, x) is continuously differentiable in x for any t ∈ R and satisfies sup (t,x)∈R×R At this point, we mention that the regularity of pulsating fronts for nonlocal KPP equations in the space periodic case was treated in [11] (see also [29]). We remark that Theorem 1.1(iii) follows directly from Proposition 1.2. We point out that the existence of transition fronts in Theorem 1.1 is proven constructively via the construction of appropriate sub-and super-solutions. The κ 0 in Theorem 1.1 is obtained in the construction of sub-solutions (see Proposition 2.3) and satisfies that κ 0 < inf t∈R κ 0 (t) (see (2.9)), where κ 0 (t) > 0 is such that The κ 0 in Theorem 1.1 may be small and hence the set of transition fronts obtained in Theorem 1.1 may only contain those which move sufficiently fast to the right. In [19], the authors proved the existence of periodic traveling waves in the time periodic case f (t+T, u) = f (t, u) also constructively via the construction of appropriate sub-and super-solutions. The κ 0 obtained in [19] is given by The κ 0 in (1.4) for the time periodic case is optimal and the value in (1.4) is nothing but the minimal speed of periodic traveling waves. But the method to construct sub-solutions in the time periodic case adopted in [19] is difficult to be applied to the general time dependent case. We adopt in the present paper a method based on an idea from [33] (also see [16,30]), which is different from that in [19] and allows us to apply it to the general time dependent case, but does not enable us to obtain the optimal value of κ 0 . It would be interesting to determine the optimal value for κ 0 (see Subsection 2.3 for some remarks).
We also remark that if a(t) = f u (t, 0) is uniquely ergodic, that is, the hull of a(t) is compact and the dynamical system generated by the shift operators on the hull of a(t) is uniquely ergodic, then the limit lim t→∞ 1 t t 0 a(s)ds exists, and hence, the asymptotic speed exists. Since interface location functions are unique up to addition by bounded functions as mentioned before, asymptotic speed (if exists) is independent of the choice of interface location functions. Note that asymptotic speed hardly exists in general.
In the presence of space regularity, i.e., Theorem 1.1(iii), we then move to the study of the asymptotic stability of u κ (t, x). To do so, we further assume the uniform exponential stability of the constant solution 1, that is, (H3) There exists θ 1 ∈ (0, 1) and Note that (H3) improves the corresponding assumption in (H2). From classical semigroup theory and comparison principle, we know that for any u 0 ∈ C b unif (R), the space of realvalued, bounded and uniformly continuous functions on R, the solution u(t, x; t 0 , u 0 ) of (1.1) with initial data u(t 0 , ·; t 0 , u 0 ) = u 0 exists globally in the space C b unif (R) and is unique. We then show for some t 0 ∈ R. Then, there holds the limit e −κx =λ > 0, then, using Theorem 1.1(ii) and the facts that X κ (t) is continuous, increasing and satisfies lim t→±∞ X κ (t) = ±∞, there exists a uniquet 0 ∈ R such that the limit lim x→∞ u 0 (x) u κ (t 0 ,x) = 1 exists, which leads to the asymptotic dynamics of u(t, x;t 0 , u 0 ) as in (1.5) with t 0 replaced byt 0 . See Corollary 4.5 for more details. In particular, if a(t) is uniquely ergodic, then u(t, x;t 0 , u 0 ) has asymptotic spreading properties in the following sense: if lim t→∞ X κ (t) t = c κ * , then for any ǫ > 0 there holds lim We remark that results as in Theorem 1.1 and Theorem 1.3 can also be established for the following reaction-diffusion equation (1.6) In particular, we have Theorem 1.4. Assume (H2) and (H3). There exists κ 0 > 0 such that for any κ ∈ (0, κ 0 ], there is a transition front u κ (t, x) of (1.6) with interface location function and satisfying the following properties It should be pointed out that transition fronts of (1.6) were established in [17], while no result concerning the stability exists in the literature. In the case f (t, u) being uniquely ergodic in t, the existence, stability and uniqueness of transition fronts of (1.6) were studied in [22].
Finally, we remark that while transition fronts of reaction-diffusion equations and nonlocal equations of Fisher-KPP type in general time or space heterogeneous media have been studied (see e.g. [16,17,18,30,33]), there exists no result in the literature concerning the corresponding discrete equations in general time or space heterogeneous media, i.e., where f (t, u) and f (i, u) are of Fisher-KPP type. Such discrete equations also arises naturally in applications (see e.g. [26]), and hence, it is of great importance to study them. We refer the reader to [6,7,8,12,15,32,34] and references therein for works of (1.7) or (1.8) in homogeneous media, i.e., f (t, u) = f (u) or f (i, u) = f (u), and to [13,14] for works of (1.8) in periodic media, i.e., f (i, u) = f (i + L, u) for some L ∈ Z. The rest of the paper is organized as follows. In Section 2, we construct appropriate global-in-time sub-and super-solutions of (1.1) for the use to construct transition fronts. In Section 3, we construct transition fronts and prove Theorem 1.1. In Section 4, we study the stability of transition fronts constructed in Theorem 1.1 and prove Theorem 1.3.

Construction of sub-and super-solutions
In this section, we construct appropriate global-in-time sub-and super-solutions of the equation (1.1). Throughout this section, we assume (H1) and (H2).

Construction of the super-solution.
For κ > 0, let The proposition then follows from f (t, u) ≤ a(t)u for u ≥ 0 by (H2) (note it is safe to extend f (t, u) to u ∈ (1, ∞) in this way).
By the second-order Taylor expansion, i.e., for some ζ x,y,t between φ κ (t, x − y) and φ κ (t, x), we see For the integral in the above equality, we first see from the monotonicity of ln and the explicit expression of φ κ (t, x) that It then follows from Lemma 2. Hence, which leads to Thus, due to inf t∈R a(t) > 0 by (H2), there is a unique κ 0 > 0 such that where we used (2.5) with s = φ κ (t, x) and α = 3 4 . The proposition then follows from a(t)g(u) ≤ f (t, u) for u ∈ [0, 1] by (H2).

Some remarks.
We make some remarks about the sub-and super-solutions constructed in the above two subsections.
x) for all (t, x) ∈ R × R (actually, the strict inequality holds). This order relation between ψ κ (t, x) and φ κ (t, x) is important in establishing various properties of approximating solutions, which will be studied in the next section, Section 3. Moreover, ψ κ (t, x) and φ κ (t, x) propagate to the right with the same speedċ κ (t). This fact says that any global-in-time solution of (1.1) between ψ κ (t, x) and min{1, φ κ (t, x)} is a transition front.
Since q 1 (κ) is continuous and increasing, we conclude (2.9). We will use (2.9) in the proof of Lemma 4.3, which is the key to the stability of transition fronts.
(iii) A possible way to enlarge the value of κ 0 is to make α change (we have fixed α = 3 4 in the above analysis). But, it seems not enough to push κ 0 arbitrary close to inf t∈R κ 0 (t). Also, inf t∈R κ 0 (t) is hardly the optimal value for κ 0 , since in the periodic case, κ 0 is exactly such that R J(y)e κ 0 y dy − 1 +â κ 0 = min κ>0 R J(y)e κy dy − 1 +â κ whereâ = 1 T T 0 a(t)dt and T is the period (see e.g. [19]).

Construction of transition fronts
In this section, we construct transition fronts and study their space regularity and decaying properties, that is, we are going to prove Theorem 1.1. Throughout this section, we assume (H1) and (H2).
Fix any κ ∈ (0, κ 0 ], where κ 0 > 0 is given in Proposition 2.3. For this fixed κ, we write The proof of the existence of transition fronts is constructive. We first construct approximating solutions. For n ≥ 1, let u n (t, x), t ≥ −n be the unique solution of (3.1) We prove some basic properties of u n (t, x) in the following Lemma 3.1. The following statements hold: Proof. (i) By Lemma 2.2(i), we have h(s) ≤ s for all s ≥ 0, which implies that u(−n, x) ≤ φ(−n, x). The result then follows from comparison principle.
The sequence {u n (t, x)} is the approximating solutions. However, we can not conclude immediately the convergence of {u n (t, x)} to a global-in-time solution of (1.1) due to the lack of regularity. Following the arguments in [16], we can show the uniform Lipschitz continuity of {u n (t, x)} in x, which of course ensures the convergence. Here, we take a different approach, which is based on the monotonicity of {u n (t, x)} as in Lemma 3.1(iii). Now, we prove Theorem 1.1.
Proof of Theorem 1.1. We first construct transition fronts. By Lemma 3.1(iii), for any fixed (t, x) ∈ R × R, there exists n 0 = n 0 (t) ≥ 1 such that the sequence {u n (t, x)} n≥n 0 is nondecreasing. Since it is clearly between 0 and 1, the limit lim n→∞ u n (t, x) exists and equals to some number in [0, 1]. Hence, there exists a function u : Moreover, since u n (t, x) satisfies u n t = J * u n − u n + f (t, u n ), we have that for any t > −n and x ∈ R, Passing to the limit n → ∞ in (3.3), we conclude from the dominated convergence theorem that for any t ∈ R and x ∈ R, From which, we conclude that u(t, x) is differentiable in t and satisfies (1.1). To see u(t, x) is a transition front of (1.1), we notice ψ(t, x) ≤ u(t, x) ≤ φ(t, x) by Lemma 3.1(i) and (3.2). Taking X(t) = c(t), we find e −κx = h ′ (0) = 1 due to Lemma 2.2(i), we find that the limit lim x→∞ u(t,x+X(t)) e −κx = 1 exists and is uniform in t ∈ R. We also have lim x→∞ u(t, x + X(t)) h(e −κx ) = 1 uniformly in t ∈ R.
(3.5) (iii) By (3.2) and Lemma 3.1(ii), we have  We are going to show v(t, x + X(t)) → 0 as x → −∞ uniformly in t ∈ R. To do so, we let ǫ > 0, and choose T = T (ǫ) > 0 and L = L(ǫ) > 0 such that where C := sup (t,x)∈R×R |u x (t, x)| < ∞ by (iv). Notice such a L exists due to (H1). For θ 1 as in (H2), let X θ 1 (t) be the interface location function at θ 1 , i.e., u(t, X θ 1 (t)) = θ 1 for all t ∈ R. It is well-defined due to the monotonicity of u(t, x) in x. Since u(t, x) is a transition front, we have sup t∈R |X θ 1 (t) − X(t)| < ∞. Setting It then follows from (3.7) that for x − X(t) ≤ −C T there holds Since J ′ is odd and u(τ, x + X(τ )) → 1 as x → −∞ uniformly in τ ∈ R, we find some We then deduce from (3.8) and (3.9) that for x − X(t) ≤ −C T,L there holds Observing the above analysis is uniform in t ∈ R, we find the limit.

Stability of transition fronts
Let κ 0 be as in Proposition 2.3 and fix κ ∈ (0, κ 0 ]. Let u(t, x) = u κ (t, x) be the transition front constructed in Theorem 1.1 with interface location function X(t) = c κ (t). We study the asymptotic stability of u(t, x). Throughout this section, we assume (H1)-(H3).
To prove Proposition 4.1, we need the uniform steepness of u(t, x) given in the following Note that by the space homogeneity of the equation (1.1), we may assume, without loss of generality, that ξ = 0. Hence, we assume where ǫ ∈ (0, ǫ * ], ω ∈ (0, ω * ] and C > 0 (to be chosen). With By Theorem 1.1(v), we find some M 0 > 0 such that By Lemma 4.2, we have C M := sup t∈R sup |x−X(t)|≤M u x (t, x) < 0. Then, we choose C = C(ǫ * , ω) > 0 such that Then, the choice of C leads to Hence, we have show N [u − ] ≤ 0, that is, u − is a sub-solution.

4.2.
Proof of Theorem 1.3. This whole subsection is devoted to the proof of Theorem 1.3. Let u 0 be as in the statement of Theorem 1.3 and u(t, x; t 0 ) := u(t, x; t 0 , u 0 ) be the solution of (1.1) with initial data u(t 0 , ·; t 0 ) = u 0 . We are going to show lim t→∞ sup x∈R Here, we only prove (4.1); the proof of (4.2) can be done along the same line except one that we comment after the proof of (4.1).
for all t ≥ t 0 . Equivalently, lim x→∞ u(t, x + X(t); t 0 ) u(t, x + X(t)) = 1 uniformly in t ≥ t 0 (4.6) To show (4.1), we actually only need the upper bound in (4.5); the lower bound in (4.5) and the limit (4.6) are needed for proving (4.2). The proof of Lemma 4.3 is postponed to Subsection 4.3 Now, for contradiction, suppose (4.4) is false, that is, ξ inf > 0. We are going to find a number in Ξ, but smaller than ξ inf .
). Arguing as in the proof of the upper bound above, we see that where we used g(v 1 )−g(v) = δv 2 g u (v * ) ≤ δv 2 , since g u ≤ 1 (note that it's safe to extend g to (−∞, 0) so that g u ≤ 1 on this interval). Since g(u) ≤ f (t, u), we find v t ≤ J * v − v + f (t, u). Then, by comparison principle, u(t, x − η; t 0 ) ≥ v(t, x) for all x ∈ R and t ≥ t 0 , that is, for t ≥ t 0 , which, together with (3.5), lead to h(e −κx ) = 1, we have lim x→∞ e −κ * x h(e −κx ) = 0. Then, a similar argument as in the proof of the upper bound gives the lower bound for u(t, x; t 0 ). This finishes the proof of (4.5).