Entire solutions in nonlocal monostable equations: Asymmetric case

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., \begin{document}$u_{t}=J*u-u+f(u)$\end{document} . Here the kernel \begin{document}$J$\end{document} is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds \begin{document}$c^{*}$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} , where \begin{document}$c^*$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel \begin{document}$J$\end{document} is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

Although the traveling wave solution is a key object characterizing the dynamics of nonlocal dispersal equations, such as (1), it is not enough to understand the whole dynamics. In fact, traveling wave solutions are special examples of the so-called entire solutions, which are defined in the whole space and for all time t ∈ R. From the viewpoint of biology, entire solutions can model new spreading and invasion behavior of the epidemic and species, respectively; see [28,30,45]. Moreover, entire solutions can help us for the mathematical understanding of transient dynamics and the structures of the global attractors. However, the global attractors are rather complicated. Some new types of entire solutions other than traveling wave solutions have been established for various evolution equations with spatially homogeneous environment; see e.g. [8,18,19,27,29,37,41] for reaction-diffusion equations with and without delays, [38] for delayed lattice differential equations with global interaction, [25] for reaction-advection-diffusion equations, [30,36,40,45] etc. for reaction-diffusion or discrete model systems.
Recently, Li et al. [26] and Sun et al. [33] constructed new types of entire solutions for symmetric nonlocal equations with monostable and bistable nonlinearity by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. And Dong et al. [16], Li et al. [28] and Zhang et al. [45] further considered the entire solutions for symmetric nonlocal systems. However, the issue of the existence of entire solutions for nonlocal equation (1) is still open when J is asymmetric. As for entire solutions, it is natural to ask what is the difference between symmetric equations and asymmetric equations.
In fact, there is a close relationship between the nonlocal equation (1) and a local version. Let J(x) = 1 ε P ( x ε ) with ε > 0, where P (x) is a general mollification function with support x ∈ [−1, 1]. If u(x) is smooth, then the Taylor's formula implies that where α = 1 2 R P (−z)z 2 dz, β = R P (−z)zdz. Thus there is a formal analogy between J * u − u and ε 2 αu + εβu (see [12]). When J is symmetric, it is clear that β = 0, then (1) can be viewed as an approximation of the classical Laplace diffusion equation Therefore, equation (1) indeed shares many properties of equation (3). For instance, both of them have maximum principle, and stationary solutions are constants [17].
Especially, the results on the existence and some related properties of traveling wave solutions of equation (1) are similar to those of the reaction diffusion equation (3); see [3,7] for bistable nonlinearity, [4,13,31,32] and the references therein for monostable nonlinearity. However, for general asymmetric kernel J, we see from (2) that a better analogy than (3) for (1) is the following elliptic equation: Thus, there is an essential difference between symmetric and asymmetric equations, which makes the two types of equations have many different dynamical properties. For instance, Coville et al. [12] and Sun et al. [34] have showed that the minimal speed c * of asymmetric equation (1) may be nonpositive (The minimal speed c * > 0 for symmetric equation (1)). Additionally, asymmetric equations lack symmetry between decreasing and increasing traveling wave solutions. Therefore, resolving the issue of entire solutions of asymmetric equation (1) represents a main contribution of our current study.
In order to establish the entire solutions of (1), it is necessary to study the property of the minimal speeds c * andĉ * . Theorem 1.1. Assume that (J1) and (F1) hold. Then c * +ĉ * ≥ 0.
The assertion (iii) of Theorem 1.4 implies that, when c > 0,ĉ < 0, the entire solution behaves as two traveling wave solutions φ(x − ct + h 1 ) andφĉ(x +ĉt + h 2 ) moving in the same direction as t → −∞. And the decreasing wave solution φ(x − ct + h 1 ) move faster than the increasing oneφĉ(x +ĉt + h 2 ) since c > −ĉ. At last, the entire solution u − (x, t) tends to 1 as t → +∞. Remark 1.5. For any c > c * ,ĉ >ĉ * and (c,ĉ) ∈ C −+ , the equation (1) also has an entire solution u So far, we only constructed entire solutions of (1) for c > c * andĉ >ĉ * . Indeed, Coville [12] also guaranteed the existence of monotone traveling wave solutions with the critical speed c = c * (c * = 0) under conditions (J1) and (F1). If we further assume that f satisfies then the existence of entire solutions of (1) combining the traveling wave solutions with critical speeds c = c * and/orĉ =ĉ * can also be established. Theorem 1.6. Assume that (J1) and (F1)-(F2) hold. When c * ĉ * = 0, for any c ≥ c * ,ĉ ≥ĉ * and cĉ = 0, h 1 , h 2 ∈ R, k > 0 and χ 1 , Recall that in [26], we have obtained the existence and relative properties of entire solutions for symmetric equation (1) with monostable nonlinearity, that is, J satisfies R J(y)dy = 1 and J is compactly supported, and f satisfies (F1). Obviously, our Theorem 1.2 and Corollary 1.3 in this paper can completely cover the previous results in [26]. However, the property (iii) in Theorem 1.4 can not occur when J is symmetric. In fact, when J is symmetric, 2 ,φĉ(0) = 1 2 and the uniqueness of traveling wave solutions. However, in [26] we did not consider the uniqueness and continuous dependence of the entire solutions on parameters c,ĉ, h i (i = 1, 2) and k. We will devote to this topic in this paper. Unfortunately, the method we used here depends on the symmetry of kernel J. Therefore, we just consider the uniqueness and continuous dependence of the entire solutions of (1) with symmetric J. The result can be stated as follows.
The greatest difficulty in proving the continuous dependency of the entire solution w(x, t) is that the mathematical expression of the solution of Cauchy problem of (1) is too abstract, since the kernel J is abstract. In this paper, we get over it by means of Fourier transform.
The rest of the paper is organized as follows. In Section 2, we give the existence of the solutions for Cauchy problem of (1) and a comparison theorem which is essential in getting the entire solutions we desired. Sections 3, 4 and 5 are devoted to the proofs of Theorems 1.1, 1.2, 1.4 and 1.6, respectively. In the last section, we prove the continuous dependence on parameters and the uniqueness of entire solutions obtained in [26], and end this paper with an important remark.

2.
Preliminaries. In this section, we will make some preparations for getting our main results later. Since the main theorems are proved by the aid of solution sequence of the Cauchy problems starting at times −n with suitable initial values, we first consider the following Cauchy problems of (1): Furthermore, if for any τ < T ,ū is a supersolution of (1) on (x, t) ∈ R × [τ, T ), then u is called a supersolution of (1) on (x, t) ∈ R × (−∞, T ). Similarly, a subsolution u(x, t) can be defined by reversing the inequality (15).
Next, we show some priori estimates uniformly in n of u n (x, t), which allow us to take the limit as n → +∞. Moreover, some properties fulfilled by the functions u n (x, t) will hold well for the limit function u(x, t).
Lemma 4.1. There exists a positive constant C 1 , which is independent of x, t, n and (χ 1 c, χ 2ĉ , χ 1 h 1 , χ 2 h 2 , χ 3 k) such that for all n ∈ N, t ≥ −n + 1 and x ∈ R, In addition, if there exists C 2 independent of n, x and (χ 1 c, χ 2ĉ , Then there exist positive constants M and M , which are independent of x, t, n and (χ 1 c, χ 2ĉ , χ 1 h 1 , χ 2 h 2 , χ 3 k), such that the solutions u n (x, t) of (22) satisfy for any x ∈ R, t > −n and η > 0.
Proof. Since the functions u n are uniformly bounded and f is of C 2 , it is easy to show that (24) holds. Moreover, in view of J ∈ L 1 (R), there exists L > 0 such that Thus, the rest of the proof is similar to that of [26, Proposition 2.5].
The following lemma gives an upper bound of the functions u n (x, t), which is independent of n.
Proof. We will only prove u n (x, t) ≤ Π 1 (x, t) for all (x, t) ∈ R × [−n, +∞), since the proofs of other inequalities are similar. Without loss of generality, we assume χ 1 = 1 and set v n (x, t) = u n (x, t) − φ c (x − ct + h 1 ). Then we will compare v n (x, t) with the solution of a linear equation. Obviously, 0 ≤ v n (x, t) ≤ 1 for (x, t) ∈ R × [−n, +∞). Since φ c (x − ct + h 1 ) is a solution of (1) and f (s) ≤ f (0) for any s ∈ [0, 1], a direct computation shows that which implies that w n (x, t) is a solution of the following Cauchy problem Note thatφĉ(ξ) ≤ Bĉe µ(ĉ)ξ for all ξ ∈ R. Thus we have v n (x, −n)

It then follows from Lemma 2.4 that
The proof is complete.
Proof of Theorem 1.2. (i) Note that φ c (x−ct) andφĉ(x+ĉt) are monotone traveling wave solutions of (1) which satisfy (5) and (6), respectively. Then for any c > c * ,ĉ > c * and cĉ = 0, we have |φ c | ≤ 2+M3 |c| and |φ ĉ | ≤ 2+M3 |ĉ| , where M 3 = max s∈[0,1] f (s), which make (25) of Lemma 4.1 hold. Thus the solutions u n (x, t) of (22) are globally Lipschitz in x. Following (24) and Lemma 4.1, the Arzela-Ascoli Theorem and the diagonal extraction imply that there exists a subsequence {u ni } i∈N of {u n } n∈N such that u ni (x, t) converge uniformly to a function u(x, t) in T . From the equation satisfied by u n (x, t), we know that the limit function u(x, t) is an entire solution of (1). Furthermore, since f is of class C 2 , the same estimate as (24) is also hold for u(x, t). That is to say, there exists a constant C 3 , which is independent of x, t and (χ 1 c, χ 2ĉ , χ 1 h 1 , χ 2 h 2 , χ 3 k), such that |u t |, |u tt | ≤ C 3 for any (x, t) ∈ R 2 .
Therefore, we have The proof of the second inequality is similar.
Next, we prove the Corollary 1.3.

It then follows from Lemma 2.4 that
A,a sinceĉ > 0. And since f is of class C 2 , the conclusion is obvious. (ii) can be proved by the same argument as that in [26, Theorem 1.1], thus we omit the details. This completes the proof.

5.
Proof of Theorem 1.6. Let v n (x, t) be the unique solution of (22) with u n (x, t) replaced by v n (x, t), we first give an upper bound of v n (x, t).
According to Lemma 5.1, the remaining proof of Theorem 1.6 is similar to that of Theorem 1.2, and is omitted.
6. Proof of Theorem 1.7. In this section, we prove Theorem 1.7 under the conditions (J2) and (F1). Since it is a further work of [26], for convenience, we use φĉ(−x −ĉt) instead ofφĉ(x +ĉt) as the nondecreasing traveling wave solutions of equation (1). Lemma 6.1. The functions φ c (z) are continuous with respect to c ∈ (c * , +∞) in 1] f (s), f ∈ C 2 (R) and J is compactly supported. By differentiating the equation It is easy to see that there exists a constant M 5 > 0 which is independent of x and c such that If c l → c ∈ (c * , +∞), then by the unique boundedness of |φ c l (z)|, |φ c l (z)| in z ∈ R on l ∈ N and by a diagonal extraction process, there exists a subsequence c li such that φ c l i → φ in C 1 By passing to the limit c li → c , the function φ is nonincreasing in R, since φ c l are normalized in 0, it follows that φ(0) = 1 2 . Note that f is positive on (0, 1), this yields that φ(−∞) = 1 and φ(+∞) = 0. Therefore, φ is a traveling wave front of (1) with speed c. Following Carr and Chmaj [4], we have φ ≡ φ c . Therefore, the whole sequence φ c l → φ c in C 1 loc (R) as l → +∞.
We claim that φ c l → φ c0 in C 1 loc (R) as l → +∞. Indeed, since c l → c 0 as l → +∞, there exists a subsequence c li and a function φ such that φ c l i → φ in C 1 loc (R), where φ is nonincreasing and satisfies On the other hand, by [34], there exists two constants q > 1 and γ > 1, independent of c l , such that Therefore, as l → +∞, e −λ(c0)z − qe −γλ(c0)z ≤ φ(z) ≤ e −λ(c0)z + qe −γλ(c0)z for any z ∈ R, which implies that φ is not a constant and satisfies lim z→+∞ φ(z)e λ(c0)z = 1. Then it follows from [34] (or Carr and Chmaj [4]) that φ ≡ φ c0 . Consequently, the whole sequence φ c l → φ c0 in C 1 loc (R) as l → +∞. Furthermore, note that the function φ c = φ c · + ln αc λ(c) is also a solution of with lim z→+∞ φ c (z)e λ(c)z = 1. Thus, in view of the uniqueness of traveling wave solution in [4] and [34], we have φ c = φ c , that is φ c = φ c · + ln αc λ(c) . In order to prove that α c is continuous in c at c 0 , it is enough to show that φ c (0) is continuous in c at c 0 . We argue by contradiction. Assume that φ c l (0) → φ c0 (0), but α c l → α c0 for a sequence c l → c 0 . Thus without loss of generality, there exists a real ε > 0 and a subsequence c l → c 0 such that α c l ≤ α c0 − ε. Consequently, one has since the function φ c (·) is continuous in C 1 loc (R) with respect to c. Consequently, This is impossible because φ c0 is decreasing. Since φ c (z) is continuous in c at c 0 for any z ∈ R, it is obvious that φ c (0) is continuous in c at c 0 . This completes the proof.
Proof. Fix c 0 ∈ (c * , +∞) and let c l → c 0 as l → +∞ with c l > c * for each l ∈ N.
We argue by contradiction. Assume that A c l → A 0 ∈ R ∪ {∞} as l → +∞ (up to extraction of some subsequence) and Then there exists L ∈ N such that for any l > L, A c l > b. On the other hand, since α c l → α c0 ≤ A c0 and λ(c l ) → λ(c 0 ), there exists a constant z 0 > 0, independent of c l , such that e λ(c l )z φ c l (z) ≤ b for any |z| > z 0 .
loc (R) and the equicontinuity of e λ(c l )z on l, there exists L > L such that e λ(c l )z φ c l (z) ≤ b for any l > L and z ∈ [−z 0 , z 0 ]. Consequently, e λ(c l )z φ c l (z) ≤ b for any l > L and z ∈ R, which contradicts to A c l > b for any l > L. The proof is complete.
In view of the a priori estimate (24) and Lemma 4.1, there exists a function w(x, t) such that w l (x, t) → w(x, t) as l → +∞ (up to extraction of some subsequence) in the sense of T . In particular, the function w(x, t) is an entire solution of (1) and also satisfy the estimate (24) and Lemma 4.1. Since the functions ξ l (t) are uniformly bounded in C 2 (R), wa can assume that they converge in C 1 loc (R) to a function ξ(t), which is a solution of ξ = f (ξ) in R and ξ(t) → ke f (0)t as t → −∞.
Then from [26], we obtain that the function w(x, t) fulfills the following estimate Now we prove that w(x, t) ≡ w(x, t) = w c,ĉ,h1,h2,k (x, t) for any (x, t) ∈ R 2 . Remember that the functions w n (x, t) converge to the function w(x, t) in the sense of T , where w n (x, t) are solutions of the Cauchy problems (w n ) t = J * w n − w n + f (w n ), x ∈ R, t > −n, with the initial conditions w n (x, −n) = w n,0 (x) := max φ c (x + cn + h 1 ), ke −f (0)n , φĉ(−x +ĉn + h 2 ) .
It is easy to see that y n and z n satisfy as n → +∞. In fact, the formula for y n comes directly from the equality φ c (y n + cn + h 1 ) = ke −f (0)n and the asymptotic behavior of φ c given by (7), so does z n . Notice that A c e −λ(c)z ≥ φ c (z) for any z ∈ R. By (36) and the definition of (y n , z n ), we get On the other hand, denote J(ξ) as the Fourier transform of J. Under the condition (J2), by [ , it is obvious that J(ξ) is differentiable, J(ξ) ∈ L 1 (R) and J (ξ) ∈ L 2 (R). In addition, because J is compactly supported, it follows from [5] that J(ξ) ∼ 1 − Aξ 2 + o(|ξ| 2 ) and J (ξ) ∼ −ξ as ξ → 0, where A = −1/2 J (0) > 0. According to [22, Lemmas 2.1 and 2.2] (see also [1]), the fundamental solution of the following Cauchy problem is the solution of it with initial value u 0 = δ 0 , and it can be decomposed as where K t (x) = R (e t( J(ξ)−1) − e −t )e ixξ dξ satisfies K t (x) L 1 (R) ≤ 2 for any t > 0. It is easy to get that S(x, t) L 1 (R) ≤ 3 for any t > 0. Furthermore, by [5, Lemma 2.2 and Remark 2.1], for some positive constants c, δ and any t > 0. Now fix a couple (x 0 , t 0 ) ∈ R 2 . For |t 0 | < n, we can compare w − w n with a solution of the linear equation which has an initial condition at time −n as the right-hand side of the inequality of (37). Thus, we have Call I, II, and III the three terms in the right-hand side of this last equality, respectively. Consider the first integral I and write it as I = I 1 + I 2 . With the change of variable z = x 0 − y, we have and cλ(c) > f (0). We have y n → −∞ as n → +∞. Then S(x 0 − y, t 0 + n)e λ(ĉ)y−λ(ĉ)h2 e (f (0)−ĉλ(ĉ))n dy → 0, since y n → −∞ and e (f (0)−ĉλ(ĉ))n → 0 as n → +∞. Thus, I → 0 as n → +∞. Similarly, we have III → 0 as n → +∞. Lastly, the integral II can be divided into three terms II 1 , II 2 and II 3 with obvious notation. First of all, zn yn S(x 0 − y, t 0 + n)|ξ(−n) − ke −f (0)n |dy ≤ 3e f (0)(t0+n) |ξ(−n) − ke −f (0)n | → 0 as n → +∞, according to S(x, t) L 1 (R) ≤ 3 for any t > 0 and ξ(t) ∼ ke f (0)t as t → −∞. Now, we deal with term II 2 . We call II 2,1 and II 2,2 the two terms on the right-hand side of the last inequality. Thus, Since K t0+n L ∞ (R) ≤ c(t 0 + n)e −δ(t0+n) J L 1 (R) +c(1 + t 0 + n) − 1 2 , we have K t0+n L ∞ (R) → 0 as n → +∞.
By the same estimates as above, we can show that the entire solution of (1) is unique.
Finally, we end this paper by giving a meaningful remark to demonstrate the differences caused by the decay rates of the traveling wave solutions and the spatially independent solution when J is symmetric and asymmetric. Remark 6.4. When J is symmetric, for any c,ĉ ≥ c * =ĉ * , we have cλ(c),ĉλ(ĉ) > f (0). Let y(t) = φ c (x(t) − ct + h 1 ) = φĉ(−x(t) −ĉt + h 2 ). Then which implies that y(t) decays faster than ξ(t) at the points x(t) as t → −∞. However, when J is asymmetric, (38) may not hold, which means the function ξ(t) may not play a part in the construction of entire solutions in Theorems 1.2, 1.4 and 1.6 even if χ 3 = 1.