Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 84, pp. 1–23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SPATIAL DYNAMICS OF A NONLOCAL BISTABLE REACTION DIFFUSION EQUATION BANG-SHENG HAN, MENG-XUE CHANG, YINGHUI YANG Abstract. This article concerns a nonlocal bistable reaction-diffusion equa- tion with an integral term. By using Leray-Schauder degree theory, the shift functions and Harnack inequality, we prove the existence of a traveling wave solution connecting 0 to an unknown positive steady state when the support of the integral is not small. Furthermore, for a specific kernel function, the stability of positive equilibrium is studied and some numerical simulations are given to show that the unknown positive steady state may be a periodic steady state. Finally, we demonstrate the periodic steady state indeed exists, using a center manifold theorem. 1. Introduction In this article, we consider the integro-differential equation ∂u ∂t = ∂2u ∂x2 + ku2(1− φ ∗ u)− bu in R× (0,∞), (1.1) where (φ ∗ u)(x) := ∫ R φ(x− y)u(y, t)dy, and φ(x) satisfies φ(x) ≥ 0, φ(0) > 0, φ(x) = φ(−x), ∫ R φ(x)dx = 1, ∫ R x2φ(x)dx <∞. (1.2) Such equations describe the nonlocal consumption of resource in the population dynamics, and in which u(≥ 0) means the density of the population (see [4, 16]). The terms in (1.1) are interpreted as follows: the term ku2 (k > 0), represents the reproduction of the population, which is in direct proportion to the density square under the sexual case, and to the available resources; the integral term describes nonlocal consumption of the resources; the term −bu corresponds to mortality of the population. Similar equations also arise in species evolution [4], ecology [15, 19], adaptive dynamics [16] (see also [9, 10, 17]), and Brownian motion [31]. 2010 Mathematics Subject Classification. 35C07, 35B40, 35K57, 92D25. Key words and phrases. Reaction-diffusion equation; traveling waves; numerical simulation; critical exponent. c©2020 Texas State University. Submitted October 31, 2019. Published July 30, 2020. 1 2 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 If the kernel φ tends to be a δ-function, equation (1.1) becomes the classical reaction-diffusion equation ∂u ∂t = ∂2u ∂x2 + ku2(1− u)− bu in R× (0,∞). (1.3) Moreover, if k2 − 4kb > 0, equation (1.3) is bistable and the nonlinearity f(u) = ku2(1− u)− bu has three zeros: u+ = 0, u0 = k − √ k2 − 4kb 2k , u− = k + √ k2 − 4kb 2k . In this case, we know (1.3) has a globally asymptotically stable traveling wave solution of the form u(x, t) = u(x− ct) with the limits u→ u± as x→ ±∞, where the constant c denotes the wave speed and is unique up to translations in space (see e.g. [28] and there references therein). Recently, much attention was devoted to introducing a nonlocal effect into the nonlinear reaction term (in fact, now researches about the nonlocality mainly focus on the monostable case, see [1, 2, 5, 6, 14, 13, 18, 23, 24, 27], the results for the bistable case are relatively seldom). When the support of the function φ is suffi- ciently small, Apreutesei et al [4] explored the property of the Fredholm operator Lu = u′′ + cu′ + [2kw(1− φ ∗ w)− b]u− kw2(φ ∗ u), where w satisfies w′′ + cw′ + kw2(1− φ ∗ w)− bw = 0. And they used it to prove that (1.1) admits traveling wave solutions connecting u+ to u−. Similar results can be established by using the method of Wang et al [29], where the quasi-monotonicity conditions are needed. When the support of the integral is not small, by using a topological degree for the proper Fredholm operator, Demin and Volpert [11] proved that the equation admits monotone traveling wave solution connecting 0 to 1 2 + √ 1 4 − α when the nonlinearities with the form of u(φ ∗ u)(1− u)−αu. Alfaro et al [3] researched the case of the nonlinearities with the form of u(u−θ)(1−φ∗u). For more results about bistable reaction-diffusion equation can be referred to [8, 25, 26, 30]. It should be pointed out that the difficulties caused by different nonlinear term (the integral located at different place) are different. And the methods used to overcome these difficulties are also different. More recently, the research about (1.1) has made some progresses. By using sub- and super-solutions for an appropriate monotone operator and cut-off approx- imation, Li et al [22] proved that equation (1.1) exists monotone traveling wave solution for the monostable case. However, there is no result for the bistable case when the support of φ is not small. The purpose of this paper is to find (at least partially) the traveling wave solution of (1.1) for the bistable case by developing the methods of Alfaro et al [3], Apreutesei et al [4] and Han et al [20]. In contrast with [3, 4, 20], the main difficulty in the study of (1.1) is to get a priori estimate of wave speed. To overcome such difficulty, we study an evolution equation and obtain some estimations. Furthermore, using the Leray-Schauder degree theory, we prove that equation (1.1) exists the traveling wave solution connecting 0 to an unknown positive steady state. In addition, in order to more clearly describe the behavior of the solution about (1.1), we give the stability analysis and numerical simulations. Now, we state our main results. EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 3 Theorem 1.1. There exists a traveling wave solution (c, u) such that − cu′ = u′′ + ku2(1− φ ∗ u)− bu in R, (1.4) with the boundary conditions lim x→+∞ u(x) = 0, lim inf x→−∞ u(x) > 0. (1.5) In above theorem, we prove that (1.1) admits traveling wave solutions connecting 0 to an unknown positive state, while we do not consider the relationship between u0 and the value of this unknown positive state at the negative infinite. In fact, our main purpose is to study whether the unknown steady state can be periodic, so the results of Theorem 1.1 can be used in future work. In addition, the numerical result about (1.1) is shown in [4] when the support of φ is not small. For completeness, we give the result when the support of the integral is small. Remark 1.2 ([4, Theorem 4.1]). Assume that φ satisfies (1.2). Then there exists ε0 > 0 such that (1.1) admits a solution (uε, cε) ∈ C2+α(R)×R satisfying (1.4) for any |ε| < ε0, and with the following boundary conditions lim x→+∞ uε = 0, lim x→−∞ uε = u−. Next we study the stability of u = u− when the kernel φ takes some special forms. Theorem 1.3. (i) If the kernel has the form φ(x) = 1 2e −|x|, then (4.3) has Turing bifurcation around (u, v) = (k+ √ k2−4kb 2k , k+ √ k2−4kb 2k ) at b = bc, where bc is defined in (4.7). (ii) If the kernel has the form φ(x) = Ae−a|x| − e−|x|, then (4.11) has Turing bifurcation around (u, v, w) = (k+ √ k2−4kb 2k , 3k+ √ k2−4kb 2k ,−2k+ √ k2−4kb 2k ) at b = bc, where bc is defined in (4.15). In Theorem 1.3, we use the kernel with two specific forms, and use the linear stability analysis to take into account the stability of the state u = u−. Moreover, when u(x, 0) takes the form of (4.8), through numerical simulation, we show that the wave can connect 0 to a periodic steady state. Next, we show that (1.1) in- deed admits periodic steady state. Previously, we gave some other assumptions on the kernel φ. After linearizing equation (1.1) around u = u−, we can obtain the dispersion relation d(λ, σ, k, b) := −σ2 + b− ku2 −φ̂(σ)− λ. (1.6) For φ(x) satisfying (1.2), we also assume that there exists a unique σc > 0, kc > 0 and bc > 0 so that (i) d(0, σc, kc, bc) = 0. (ii) ∂σd(0, σc, kc, bc) = 0. (iii) ∂σσd(0, σc, kc, bc) < 0. Then, the result about the existence of stationary periodic solutions of (1.1) can be stated as follows. Theorem 1.4. Assume that φ(x) satisfies (1.2) and d(λ, σ, k, b) satisfies the three conditions (i)-(iii) above. Let k = kc + ε2, b = bc + ε2 2 and σ = σc + δ. Then there 4 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 exists an ε0 > 0 such that equation (1.1) has a 2π/σ-periodic solution with leading expansion of the form uε,δ(x) = u− + √ Λ cos((σc + δ)x) +O(|Q|) for all ε ∈ (0, ε0] and δ satisfying( − kc − 2bc + √ k2 c − 4kcbc 4 φ̂′′(kc)− 1 ) δ2 < (4bc + ε2)φ̂(σc + δ)√ k2 c − 4kcbc + √ k2 c − 4kcbc − 4bcε2 − ε4 ε2 + ε2 2 , where Λ is defined in (5.5) and Q is defined in (5.7). This article is organized as follows. In Section 2 and 3, we prove Theorem 1.1. In Section 4, we research the stability of the state u = u−, that is Theorem 1.3. In Section 5, we give the proof of Theorem 1.4. Finally, further discussions are made in Section 6. 2. Existence of traveling wave solutions In this section, we construct a traveling wave solution of (1.1). Specifically, in subsection 2.1, we use the method of [3, 7] to give a priori estimates of solution u in a finite domain. In subsection 2.2, we construct a solution (c, u). 2.1. A priori estimates of solution u in a finite domain. For a > 0 and 0 ≤ τ ≤ 1, we seek a function u = uaτ ∈ C2([−a, a],R) and a speed c = caτ ∈ R satisfying Tτ (a) : { −u′′ − cu′ = ku(u− u0)(u− − u) + τku2(u− φ ∗ û) in(−a, a), u(−a) = u−, u(0) = ε/2, u(a) = 0, (2.1) where û =  u−, in (−∞,−a), u, in (−a, a), u+, in (a,∞), and the number ε will be determined later. Firstly we introduce a homotopy from T0(a) (a local problem) to Tτ (1) (a nonlocal problem). Secondly, we obtain a solution of T1(a) by using a Leray-Schauder degree. For convenience, in the sequel, we often replace u with û. If u(xl) = min x∈[−a,a] u(x) and −u′′ − cu′ = ku(u − u0)(u− − u) on a neighborhood of xl, we obtain u ≡ ul by the maximum principle. But u ≡ ul is obviously impossible. So any solution of Tτ (a) satisfies u ≥ 0, and by the maximum principle we obtainI cahanged Pτ (a) to Tτ (a). Please check it u > 0, −u′′ − cu′ = ku(u− u0)(u− − u) + τku2(u− φ ∗ u) in (−a, a). (2.2) The following lemma gives a priori bounds for u. Lemma 2.1. There exist M(φ) > u− and a0 > 0, such that for every 0 ≤ τ ≤ 1 and a ≥ a0, any solution of Tτ (a) satisfies 0 ≤ u(x) ≤M, ∀x ∈ [−a, a]. EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 5 Proof. If τ = 0, that is T0(a), −u′′ − cu′ = ku(u− u0)(u− − u) in (−a, a), u(−a) = u−, u(0) = ε 2 , u(a) = 0, we can obtain directly 0 ≤ u(x) ≤ u− ≤ M . Now, for 0 < τ ≤ 1, assume M := maxx∈[−a,a] u(x) > u− (otherwise, the conclusion is trued). Since u(−a) = u− and u(a) = 0, there exists a point xm ∈ (−a, a), such that u(xm) = M . By evaluating (2.2) at xm, we obtain (φ∗u)(xm) ≤ kM−b kM < 1. In addition, from u ≥ 0, we obtain −u′′ − cu′ = ku(u− u0)(u− − u) + τku2(u− φ ∗ u) ≤ ku2 ≤ kM2 . (2.3) Firstly, we consider the case of c < 0. Multiplying e−|c|z on (2.3) and integrating from x to xm yields∫ xm x (u′(z)e−|c|z)′dz ≥ − ∫ xm x kM 2 e−|c|zdz. Since u′(xm) = 0, separating u′(x) and integrating from x to xm, we obtain∫ xm x u′(z)dz ≤ −kM 2 |c| ∫ xm x (e−|c|(xm−z) − 1)dz. According to u(xm) = M and separating u(x), we have u(x) ≥M [1− kM(x− xm)2g(|c|(xm − x))], where g(y) := e−y+y−1 y2 . It is clear that g(y) ≤ 1/2 for y > 0, which implies u(x) ≥M [ 1− kM 2 (x− xm)2 ] ∀x ∈ [−a, xm]. (2.4) From u(−a) = u− it follows that 1 > u− ≥M [ 1− kM 2 (a+ xm)2 ] . (2.5) Now take a0 = 1/ √ kM and let x0 := 1√ kM . If xm ∈ (−a,−a + x0), inequality (2.5) shows that M ≤ (1 − 1 2 )−1 = 2. If xm ∈ [−a+ x0, a), using (2.4) yields 1 ≥ (φ ∗ u)(xm) ≥ ∫ x0 0 φ(z)u(xm − z)dz ≥M ∫ x0 0 φ(z) ( 1− kM 2 z2 ) dz. From the definition of x0, we obtain 1 ≥ M 2 ∫ 1/ √ kM 0 φ(z)dz ≥ M 2 ∫ 1/ √ kM 0 (φ(0)− ‖φ′‖L∞(−1,1)z)dz, which implies M ≤ (4k + ‖φ′‖)2 4kφ2(0) . 6 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 Choose M = (4k + ‖φ′‖)2/(4kφ2(0)) and then we can complete the proof of the case c < 0. The case c > 0 can be proved in a similar way by integrating on [xm, a] rather than on [−a, xm]. Lastly if c = 0, by integrating twice the inequality −u′′ ≤ kM2 on [x, xm], we immediately achieve (2.4). The rest of the process is similar to the above. This completes the proof. � Next, we show a priori estimates for c. Lemma 2.2. (i) For each ε ∈ (0, u−4 ), there exists a0(ε) > 0 such that, for all 0 ≤ τ ≤ 1 and a ≥ a0, any solution of Tτ (a) satisfies c ≤ 2 √ kM =: cmax, where M is defined in Lemma 2.1. (ii) For any a > 0, there exists ĉmin(a) > 0, such that, for all 0 ≤ τ ≤ 1, any solution (c, u) of Tτ (a) satisfies c ≥ −ĉmin(a). (iii) There exists cmin > 0 and a0 > 0, such that, for all a ≥ a0, any solution (c, u) of T1(a) satisfies c ≥ −cmin. Proof. (i) Note that u(≥ 0) satisfies − u′′ − cu′ = ku(u− u0)(u− − u) + τku2(u− φ ∗ u) ≤ ku2 ≤ kMu. (2.6) We use the contrapositive method, by assuming that c > 2 √ kM . We define ϕA(x) = Ae− √ kMx which satisfies − cϕ′A − ϕ′′A > kMϕA. (2.7) Because of u(x) ∈ L∞(−a, a), we have u(x) < ϕA(x) when A > 0 is sufficiently large and u(x) > ϕA(x) when A < 0. Then, we can define A0 = inf{A : ϕA(x) > u(x) for all x ∈ [−a, a]}. Obviously, there exists x0 ∈ [−a, a] such that ϕA0 (x0) = u(x0) and A0 > 0. Using (2.6), (2.7) and the maximum principle, we know that x0 /∈ (−a, a). Because A0 > 0, we have x0 = −a. Combining this with ϕA0 (−a) = u−, we have A0 = u−e − √ kMa. However, u(0) ≤ ϕA0(0) = u−e − √ kMa < u(0) = ε/2 when a > 1√ kM (ln 2u− − ln ε), there must be c ≤ 2 √ kM . Choose a0 ≥ 1√ kM (ln 2u−− ln ε) and then we completes the proof of (i). (ii) Giving a > 0, the solution of Tτ (a) satisfies −u′′ − cu′ + (M2 + 1)u ≥ 0, and u(−a) = u−, u(a) = 0. In view of M2 + 1 ≥ 0, by the comparison principle we know that u ≥ v, where v satisfies −v′′ − cv′ + (M2 + 1)v = 0, v(−a) = u−, v(a) = 0. From precise calculation, v(x) = u− e−λ++a − e(λ+−2λ−)a eλ +x − e(λ+−λ−)a e−λ++a − e(λ+−2λ−)a eλ −x, where λ± = −c± √ c2+4(M2+1) 2 . We see that v(0) → u− as c → −∞. Thus, for any a > 0, there exists ĉmin(a) > 0 such that c ≤ −ĉmin(a) which implies ε/2 < v(0) ≤ u(0) and leads u not to be the solution of Tτ (a). Thus, any solution (c, u) of Tτ (a) with 0 ≤ τ ≤ 1 requires c ≥ −ĉmin(a). This proves (ii). EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 7 (iii) To obtain a priori lower bound for c of the equation ∂2u ∂x2 + c ∂u ∂x + ku2(1− φ ∗ u)− bu = 0, x ∈ (−a, a), we take into account the evolution equation ∂u ∂t = ∂2u ∂x2 + c ∂u ∂x + ku2(1− φ ∗ u)− bu. (2.8) So the solution we need is a stationary solution of (2.8). Take a solution u(x, t) of equation (2.8) and let v = u − u0, where u0 = k− √ k2−4kb 2k < 1/2 and u0 satisfies u0(1− u0) = b/k. Then v satisfies ∂v ∂t = ∂2v ∂x2 + c ∂v ∂x + k(v + u0)2(1− (φ ∗ v)− u0)− bu. Suppose that v < 0 and 0 < u < 1. Then k(v + u0)2(1− (φ ∗ v)− u0)− bu = ku(v + u0)(1− (φ ∗ v)− u0)− ku0(1− u0)u < −k(φ ∗ v). (2.9) Next, we analyze the equation ∂z ∂t = ∂2z ∂x2 + c ∂z ∂x − k(ζ ∗ z), (2.10) where (ζ ∗ z)(x) = ∫ R ζ(x− y)z(y, t)dy, and ζ(x) is a piecewise constant function. That is to say ζ(x) = { M = supx φ(x), if x ∈ [−N,N ], 0, otherwise. where [−N,N ] is the support of the function ζ(x). Next we seek a solution of the equation χ′′ + c0χ ′ − k(ζ ∗ χ) = 0, (2.11) where c0 may be different from c and χ(x) has an exponential form, such as χ(x) = −eλx, then λ2 + c0λ− kM λ (eλN − e−λN ) = 0. For all M and N , this equation has a solution λ if c0 is sufficiently small. Choosing this values of λ and c0 and researching the corresponding solution ζ(x) of (2.10), then z(x, t) = χ(x − (c0 − c)t) satisfies (2.9), which has a constant outline and transfer to the right with the speed c0−c. On the other hand, the function ũ(x, t) = z(x, t) + w0 satisfies ∂ũ ∂t = ∂2ũ ∂x2 + c ∂ũ ∂x − k(ζ ∗ (ũ− w0)). (2.12) Now let us compare the solution u(x, t) of (2.8) with ũ(x, t) of (2.11), we know that u(x, t)→ u± as x→ ±∞ for all t ≥ 0 and ũ(x, t) is strictly decreasing, converging to w0 as x→ −∞ and exponentially growing at −∞. 8 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 Thus we choose a constant h such that, for all x ∈ R, u(x, 0) < ũ(x−h, 0). After that, we prove that u(x, t) < ũ(x − h, t) for all x ∈ R and t ≥ 0. Assume it does not hold, then there exists t0 > 0 such that u(x, t0) ≥ ũ(x− h, t0) for all x ∈ R, u(x0, t0) ≥ ũ(x0 − h, t0) for some x0 ∈ R. From ũ(x0 − h, t0) < u0, it follows that v(x0, t0) < 0, which holds on some neigh- borhood δ(x) of x = x0. Moreover, using 0 < u(x, t) < 1 and φ ∗ v < 0 and (2.9), we know that ku2(1− φ ∗ u)− bu < −kφ ∗ v = −kφ ∗ (u− u0) ≤ −kζ ∗ (ũ− u0), x ∈ δ(x0). which is a contradiction with (2.8) and (2.12). If there exists a stationary solution u(x) such that (2.8), then put u(x, 0) = χ(x) and obtain u(x) = u(x, t) > ũ(x, t). Thus, c0 − c < c, where c0/2 is chosen above. To calculate lower bound of the speed c, similar to the above process, we con- struct χ(x)→ w0 as x→∞ and it exponentially decrease as x→∞. It spread to the left with (a certain speed ) c− c0. This completes the proof. � Remark 2.3. Using above method, we cannot obtain a priori upper bound for c, because we can not get the specific nature of u as x→ −∞. In addition, for τ > 0 above method is true. If we discuss the case of τ = 0 again, we can get a consistent with the lower bound of cmin. 2.2. Construction of a solution (c, u). We will construct a solution (c, u) of T1(a) by using Leray-Schauder degree argument in the following Proposition. Proposition 2.4. There exists K > 0 and a0 > 0, such that, for all a ≥ a0, a solution (c, u) of T1(a), i.e. −u′′ − cu′ = ku2(1− φ ∗ u)− bu in (−a, a), u(−a) = u−, u(0) = ε 2 , u(a) = 0, u > 0, on (−a, a). (2.13) and ‖u‖C2(−a,a) ≤M, −cmin ≤ c ≤ cmax. Proof. Give v ≥ 0 defined on (−a, a) and satisfying the boundary conditions v(−a) = u− and v(a) = 0. We consider a family of linear problems Fτ (a) : { −u′′ − cu′ = ku(u− u0)(u− − u) + τkv2(v − φ ∗ v) in (−a, a), u(−a) = u−, u(a) = 0. Define Lτ : R× C1,α(−a, a)→ R× C1,α(−a, a) as Lτ : (c, v) 7→ (ε 2 − v(0) + c, ucτ := the solution of Fτ (a) ) . where the norm is ‖(c, v)‖X := max(|c|, ‖v‖C1,α). To find the nontrivial part of the kernel of Id−L1, we construct (c, u) of T1(a). Then the Leray-Schauder topological can be applied here because Lτ is compact and continuity depends on the parameter 0 ≤ τ ≤ 1. Define the set E := { (c, v) : −cmin(a)− 1 < c < cmax + 1, v > 0, ‖v‖C1,α < M + 1 } ⊂ X, EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 9 where cmin(a) and cmax are defined in Lemma 2.2, and M is defined in Lemma 2.1. From Lemmas 2.1 and 2.2, it is easy to see that there exists a0 > 0 such that, for any a ≥ a0, any 0 ≤ τ ≤ 1, the operator Id − Lτ can not vanish on the boundary ∂E. Therefore, by the homotopy invariance of the degree, we obtain deg(Id− L1, E, 0) = deg(Id− L0, E, 0). In addition, from the graph of −u′′ − cu′ = ku(u− u0)(u− − u), it follows that uc0 is decreasing with respect to c. Consequently, by using two additional homotopies (see [7, p.2834] or [3] for details), we have deg(Id− L0, E, 0) = −1, thus deg(Id− L1, E, 0) = −1 which implies that there is a solution (c, u) ∈ E of T1(a). Finally, it follows from Lemma 2.2 that cmax ≥ c ≥ −cmin. This completes the proof. � Remark 2.5 (Existence of solution on R). Equipped with the solution (c, u) of T1(a) in Proposition 2.4, now let a→∞. This enables to construct, by passing to a subsequence an →∞, a speed −cmin ≤ c∗ ≤ cmax and a function U : R 7→ (0,M) in C2 b (R) such that −U ′′ − cU ′ = kU2(1− φ ∗ U)− bU, U(0) = ε 2 . 3. Behavior of the solution at ±∞ In Section 2, we prove that there exists a traveling wave solution of (1.1) on R, in order to complete it, we also need to research the behavior of the solution at ±∞. Firstly, we rewrite equation (1.1) as −cu′ − u′′ = ku2(1− φ ∗ u)− bu = ku(u− u0)(u− − u) + ku2(u− φ ∗ u). (3.1) Secondly, we show that the solution in the box cannot attain ε/2 except that at x = 0. Proposition 3.1. For each a ≥ a0, the solution (c, u) of (2.13) satisfies u(x) = ε 2 if and only if x = 0. Proof. It follows from Proposition 2.4, that there exists a pair of (cτ , uτ ) satisfying −u′′τ − cτu′τ = kuτ (uτ − u0)(u− − uτ ) + τku2 τ (uτ − φ ∗ uτ ) in (−a, a), uτ (−a) = u−, uτ (0) = ε 2 , uτ (a) = 0, and depending continuously upon 0 ≤ τ ≤ 1. For τ = 0, the solution uτ of −u′′τ − cτu′τ = kuτ (uτ − u0)(u− − uτ ) in (−a, a), uτ (−a) = u−, uτ (0) = ε 2 , uτ (a) = 0, satisfies uτ (x) = ε/2 if and only if x = 0. Thus we can define τ∗ := sup { 0 ≤ τ ≤ 1,∀σ ∈ [0, τ ], uσ(x) = ε 2 iff x = 0 } . If it does not hold, then there exists a x∗ 6= 0 such that uτ∗(x ∗) = ε/2. Therefore, we consider the following two cases: 10 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 Case 1: x∗ < 0, and uτ∗ > ε/2 on (x∗, 0). It follows from the definition of τ∗ that uτ∗ ≥ ε/2 on (−a, 0), which implies that u′τ∗(x ∗) = 0. In addition, for the interval (x∗, 0), we consider the equation −u′′τ∗ − cτ∗u′τ∗ = kuτ∗(uτ∗ − u0)(u− − uτ∗) + τ∗ku2 τ∗(uτ∗ − φ ∗ uτ∗) ≤ 0 on (x∗, 0). When uτ∗ is small, (uτ∗ − u0)(u− − uτ∗) < 0, uτ∗ − φ ∗ uτ is bounded, and u2 τ∗ is a high-end terms. Furthermore, by the weak maximum principle, we know that uτ∗ can reach the extremum on x∗ or 0, which leads to a contradiction. Case 2: x∗ > 0 and uτ∗ < ε/2. By the definition of τ∗, one must have uτ∗ < ε/2 on (0, a) which implies that u′τ∗(x ∗) = 0. In addition, −u′′τ∗ − cτ∗u′τ∗ = kuτ∗(uτ∗ − u0)(u− − uτ∗) + τ∗ku2 τ∗(uτ∗ − φ ∗ uτ∗) ≤ 0 on (0, x∗), and u(x∗) = maxx∈(0,x∗) u(x). In view of the Hopf’s Lemma, it has u′τ∗(x ∗) > 0, which causes a contradiction. Thus uτ∗ attains ε/2 only at x = 0. The rest is to prove τ∗ = 1. By contradiction, which assuming that 0 ≤ τ∗ < 1. It follows from the definition of τ∗ that there exists a sequence (τn, xn), such that τn ↓ τ∗, xn 6= 0, and uτn(xn) = ε/2. Furthermore, take the sequence xn converging to a limit x∗ which implies that uτ∗(x ∗) = ε/2. Thus x∗ = 0 implies xn → 0. Then for some −1 ≤ cn ≤ 1, we have ε 2 = uτn(xn) = uτn(0) + u′τn(0)xn + Cn‖uτn‖C1,α |xn|1+α; that is |u′τn(0)| ≤ |Cn|‖uτn‖C1,α |xn|α ≤ C|xn|α → 0 as n→∞, By the continuity of (u′τ ) about τ , we know that u′τ (0) = 0. It has uτ > ε/2 on (−a, 0), which is a contradiction. So τ∗ = 1 and this completes the proof. � Remark 3.2. From Proposition 3.1, we know that u(x) > ε/2 on (−a, 0) and u(x) < ε/2 on (0, a). Thus let a→ +∞, we can get u(x) { ≥ ε/2, x ∈ (−∞, 0], ≤ ε/2, x ∈ [0,+∞). Lemma 3.3. For all K > 0, and α < β, there exists ε = ε(K,α, β) > 0 such that if u is a solution of (3.1) with c ∈ [α, β], 0 < u ≤ K and infx∈R u(x) > 0, then infx∈R u(x) > ε. Proof. Assume that there exists (cn, un) satisfying (3.1). Let βn := inf R un > 0, for all n, satisfying βn → 0 as n → +∞. Define vn = un βn . Since infR vn = 1 for all n, we assume that there exists a sequence xn ∈ R such that vn(xn) ≤ 1 + 1 n . Let wn = 1 vn(x+xn) satisfying −cnw ′ n w2 n − wnw ′′ n − 2w′n w2 n = kβn w2 n (1− φ ∗ ũn)− b βn wn , where ũn = u(x+ xn). Note that supx∈R un(x) ≤ K and the coefficients above are uniformly bounded (with respect to n), therefore one can extract wnk(x)→ w∞(x) (locally uniformly) and cn → c̃ ∈ [α, β]. Moreover based on ũn(0) ≤ βn(1 + 1 n ), EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 11 it follows from the Harnack inequality that ũn(x) → 0 locally uniformly in x. In addition, w∞(x) satisfies −cnw ′ ∞ w2 ∞ − w∞w ′′ ∞ − 2w′∞ w2 ∞ = − b w∞ . From the strong maximum principle together with w∞(0) = 1 and w∞(x) ≤ 1, it gets that w∞(x) ≡ 1, which is a contradiction with b 6= 0. � Lemma 3.4. Let u satisfy (3.1) with initial conditions u(0) = ε/2. Then there exists a sequence xn, such that |xn| → +∞ and u(xn)→ 0 as n→ +∞. Proof. If infR u > 0, it follows from Lemma 3.3 and u ≤ K that infR u ≥ ε , which is a contradict with u(0) = ε/2. � Proposition 3.5. Let (c, u) be the solution of T1(∞) which constructed in the end of Section 2, then lim x→+∞ u(x) = 0, and when x > 0, u(x) is monotonically decreasing to zero. Proof. Assume that the conclusion does not hold. Combining with Lemma 3.4, there exists {xn} > 0, such that u(xn) = maxx∈R u(x), then − cu′(xn)− u′′(xn) ≥ 0. (3.2) In addition, it follows from Remark 3.2 that u(x) < ε/2 when x > 0. As long as ε small enough, we can get ku(1− φ ∗ u)− b < 0, so ku2(1− φ ∗ u)− bu = u{ku(1− φ ∗ u)− b} < 0, for x > 0. (3.3) From (3.2) and (3.3), we obtain a contradiction. Thus limx→+∞ u(x) = 0 and u(x) is monotonically decreasing to zero when x > 0. � Proof of Theorem 1.1. From Remark 2.5, we know that there exists a pair of (c, u) satisfying (1.4). The remains to prove that u satisfies the condition (1.5). It follows from Proposition 3.5 that lim x→+∞ u(x) = 0. Next, we prove the behavior of the solution u(x) at negative infinity. From Remark 3.2, we know that u(x) ≥ ε/2 for all x ∈ (−∞, 0]. Then lim inf x→−∞ u(x) ≥ ε 2 > 0. So the condition (1.5) is established. This completes the proof. � 4. Linear stability analysis and numerical simulations In Section 2 and 3, we prove that (1.1) admits a traveling wave solution connect- ing 0 to an unknown positive steady state when the support of φ is not small. In this section, we mainly study the behavior of the unknown state. Firstly, we show the stability of the positive steady state by using the linear stability analysis. Sec- ondly, the special form of the solution for (1.1) is proved through using numerical simulations. For convenience, equation (1.1) can be written as ∂u ∂t = d ∂2u ∂x2 + ku2(1− (φσ ∗ u))− bu, (4.1) 12 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 for x ∈ R, t > 0, where k > 0, d > 0, b > 0. Here, we only consider two specific kernel functions (K1) and (K2). (K1) φ(x) = φσ(x) = 1 2e −|x|, where σ > 0 is a constant. Let v(t, x) = (φσ ∗ u)(t, x). (4.2) Its second order derivative with respect to x is vxx = − 1 σ2 (u− v). Then (4.1) may be replaced by ut = duxx + ku2(1− v)− bu, 0 = vxx + 1 σ2 (u− v), (4.3) for (x, t) ∈ R× (0,∞). System (4.3) has three equilibria: (0, 0),(k −√k2 − 4kb 2k , k − √ k2 − 4kb 2k ) , (k + √ k2 − 4kb 2k , k + √ k2 − 4kb 2k ) . Setting u = k+ √ k2−4kb 2k + ũ and v = k+ √ k2−4kb 2k + ṽ, substituting them into (4.3) and linearizing gives ũt = dũxx + bũ− 1 2 (k − 2b+ √ k2 − 4kb)ṽ, 0 = ṽxx + 1 σ2 (ũ− ṽ). (4.4) Firstly we consider the stability of the point (k+ √ k2−4kb 2k , k+ √ k2−4kb 2k ), which is equivalent to judging the sign of η about the characteristic equation∣∣∣∣η − b k−2b+ √ k2−4kb 2 − 1 σ2 η + 1 σ2 ∣∣∣∣ = 0; that is η2 + ( 1 σ2 − b)η + k − 4b+ √ k2 − 4kb 2σ2 = 0. (4.5) From (4.5), we know that η is negative when σ is sufficiently small, and hence (u, v) = (k+ √ k2−4kb 2k , k+ √ k2−4kb 2k ) is stable. However, η may greater than 0 as σ increasing, which implies the uniform steady state will loss the stability. Next, we consider whether the equation (4.1) will occur Hopf bifurcation or Turing bifurcation around the equilibrium point (u, v) = (k+ √ k2−4kb 2k , k+ √ k2−4kb 2k ) when this equilibrium point is unstable, that is to prove (i) of Theorem 1.3. Proof of (i) of Theorem 1.3. Take a test function of the form( ũ ṽ ) = ∞∑ k=1 ( C1 k C2 k ) eλt+ilx, (4.6) where l is real. Then substituting (4.6) into (4.5) yields∣∣∣∣b− dl2 − λ −k−2b+ √ k2−4kb 2 − 1 σ2 − 1 σ2 − l2 ∣∣∣∣ = 0. EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 13 Thus (λ− b+ dl2)( 1 σ2 + l2) + k − 2b+ √ k2 − 4kb 2σ2 = 0, which is equivalent to λ = b− dl2 − k − 2b+ √ k2 − 4kb 2 + 2σ2l2 . Note that this implies λ is real for all values of b and thus Hopf bifurcations from the uniform state (u, v) = ( k+ √ k2−4kb 2k , k+ √ k2−4kb 2k ) of the equation (4.3) are impossible. Moreover, as b increases it is possible for λ to pass through 0, indicating a loss of stability of the uniform steady state. For a fixed value of the wave number l, this occurs when( 2 σ2 + l2 )2 b2 − (4dl2 + k σ4 + 6dl4 + l2k σ2 + 2dl6 ) b+ d2l4 + kdl2 σ4 + d2l8 + 2d2l6 + dkl4 σ2 = 0. For convenience, let A = ( 2 σ2 + l2)2, B = 4dl2 + k σ4 + 6dl4 + l2k σ2 + 2dl6, C = d2l4 + kdl2 σ4 + d2l8 + 2d2l6 + dkl4 σ2 . Then b = bc := B ± √ B2 − 4C 2A , (4.7) 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 l b Figure 1. The phase shows the relation between b and l at the different value of σ with the parameter values d = 1, k = 5. The green, red, blue curve respectively represents σ = 1.2, 1, 0.8, (For interpretation of the reference to color in this figure legend, the reader is referred to the web version of this article.) For discussing the front, we mainly consider the case b = B− √ B2−4C 2A . Next, we consider the relation between b and l, we can easily know there exists a lc, such that 14 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 b has a maximum bmax = b(lc). To see the relation of l and b more clearly, in there, we take the parameter d, k, σ with specific value, and obtain the diagram about b and l (see Figure 1). Thus as b increases through bmax, the uniform steady state (u, v) = (k+ √ k2−4kb 2k , k+ √ k2−4kb 2k ) loses stability and it is anticipated that a new, non-uniform steady state will appear having a spatial structure similar to exp(ilx). This prove (i) of Theorem 1.3. � Similar to the process above, we obtain the relation between σ and l (we omit the process). Following we research the influence of σ for the solution of the equation (4.1) by using the numerical simulation. Before our numerical simulation, the initial value problem needs to be developed first. We define the initial condition of u(x, t) as u(x, 0) = { u−, for x ≤ L0, 0, for x > L0. (4.8) From (4.2), we know that v(x, 0) is determined by v(x, 0) = ∫ R 1 2σ e− |x−y| σ u(y, 0)dy; (4.9) then v(x, 0) = { u− − u− 2 e x−L0 σ , for x ≤ L0, u−e − x−L0 σ , for x > L0. (4.10) The zero-flux boundary conditions were applied here. Along with (4.8)-(4.10), system (4.4) can be simulated through the pdepe package in Matlab (see Figure 2). From Figure 2 we see that equation (4.1) admits monotone traveling wave so- lution connecting 0 to k+ √ k2−4kb 2k when σ is small. As σ increasing, the solution will occur a ’hump’, and the travelling wave loses its monotone. Moreover, as σ being much larger, the ’hump’ is being much steeper. If the value of σ is continue to increase, then the stability of the state u = k+ √ k2−4kb 2k will lose. Furthermore, a periodic steady state will replace the state u = k+ √ k2−4kb 2k . (K2) φ(x) = φσ(x) = A σ e − aσ |x| − 1 σ e − |x|σ , where A = 3a 2 > 0, a ∈ ( 2 3 , √ 2 3 ). Define v(t, x) = (A σ e− a σ |x| ∗ u ) (t, x), w(t, x) = ( 1 σ e− |x| σ ∗ u ) (t, x). and let φσv = A σ e − aσ |x|, φσw = 1 σ e − |x|σ . Then vxx = − 1 σ2 (3a2u− a2v), wxx = − 1 σ2 (−2u− w). So equation (4.1) reduces to the system ut = duxx + f1(u, v, w), 0 = vxx + f2(u, v, w), 0 = wxx + f3(u, v, w), (4.11) where f1(u, v, w) = ku2(1− v−w)− bu, f2(u, v, w) = 1 σ2 (3a2u−a2v), f3(u, v, w) = 1 σ2 (−2u− w). Obviously, systems (4.11) has three equilibria (u∗1, v ∗ 1 , w ∗ 1) = (0, 0, 0), (u∗2, v ∗ 2 , w ∗ 2) = (u0, 3u0,−2u0), (u∗3, v ∗ 3 , w ∗ 3) = (u−, 3u−,−2u−). EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 15 0 10 20 30 400 5 10 15 0 1 2 distance x σ=0.5 time t s p e c ie s u 0 10 20 30 40 0 5 10 15 0 1 2 distance x σ=1.25 time t s p e c ie s u 0 10 20 30 40 0 5 10 15 0 1 2 distance x σ=2.5 time t s p e c ie s u 0 20 40 0 5 10 15 0 0.5 1 1.5 2 distance x σ=4 time t s p e c ie s u Figure 2. Numerical simulations of the time evolution and space evolution for the bistable nonlocal equation (4.1) with kernel φσ(x) = 1 2σ e − |x|σ . The computational domain is x ∈ [0, 40], t ∈ [0, 15]. The parameter values: d = 1, k = 5, b = 1, σ are followed by 0.5, 1.25, 2.5, 4. We will mainly analyze system (4.11) to get the dynamical behavior of system (4.1). Now, linearizing system (4.11) around (u∗3, v ∗ 3 , w ∗ 3) we can get ut = duxx + a11u+ a12v + a13w, 0 = vxx + a21u+ a22v + a23w, 0 = wxx + a31u+ a32v + a33w. (4.12) where a11 = ∂f1 ∂u ∣∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = b, a12 = ∂f1 ∂v ∣∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = −u− + b, a13 = ∂f1 ∂w ∣∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = −u− + b, a21 = ∂f2 ∂u ∣∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = 3a2 σ2 , a22 = ∂f2 ∂v ∣∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = − a 2 σ2 , a23 = ∂f2 ∂w ∣∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = 0, a31 = ∂f3 ∂u ∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = − 2 σ2 , a32 = ∂f2 ∂v ∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = 0, a33 = ∂f3 ∂w ∣∣ (u∗3 ,v ∗ 3 ,w ∗ 3 ) = − 1 σ2 . 16 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 Similarly, we first consider the stability of the equilibrium point (u∗3, v ∗ 3 , w ∗ 3), which is equivalent to judge the sign of η about the characteristic equation∣∣∣∣∣∣ η − b 1 2 (k − 2b+ √ k2 − 4kb) 1 2 (k − 2b+ √ k2 − 4kb) − 3a2 σ2 η + a2 σ2 0 2 σ2 0 η + 1 σ2 ∣∣∣∣∣∣ = 0. That is (η − b) ( η + a2 σ2 )( η + 1 σ2 ) + 1 2σ2 (k − 2b+ √ k2 − 4kb) ( (3a2 − 2)η + a2 σ2 ) . (4.13) From (4.13) we know that η is negative when σ is sufficiently small, and hence (u, v, w) = ( k+ √ k2−4kb 2k , 3k+ √ k2−4kb 2k , −2k+ √ k2−4kb 2k ) is stable. However, the uni- form steady state will lose stability as σ increasing. Next, we consider whether the equation (4.1) will has Hopf bifurcation or Turing bifurcation around the equilib- rium point (u, v, w) = ( k+ √ k2−4kb 2k , 3k+ √ k2−4kb 2k , −2k+ √ k2−4kb 2k ) when this equilib- rium point is unstable. Proof (ii) of Theorem 1.3. Similar to [17], we defineuv w  = ∞∑ k=1 C1 k C2 k C3 k  eλt+ilx, (4.14) where λ is the growth rate of perturbations in time t, l is the wave speed. So, substituting equation (4.14) into equation (4.12), we can obtain detA = ∣∣∣∣∣∣ a11 − dl2 − λ a12 a13 a21 a22 − l2 a23 a31 a32 a33 − l2 ∣∣∣∣∣∣ = 0. Then ∣∣∣∣∣∣ b− dl2 − λ − 1 2 (k − 2b+ √ k2 − 4kb) − 1 2 (k − 2b+ √ k2 − 4kb) 3a2 σ2 − a2 σ2 − l2 0 − 2 σ2 0 − 1 σ2 − l2 ∣∣∣∣∣∣ = 0. which is equivalent to (b− dl2 − λ) ( a2 σ2 + l2 ) ( 1 σ2 + l2)− 3a2 2σ2 (k − 2b+ √ k2 − 4kb)( 1 σ2 + l2) + 1 2 ( k − 2b+ √ k2 − 4kb )( a2 σ2 + l2 ) 2 σ2 = 0. Note that this implies λ being real for all values b and thus Hopf bifurcation from the uniform state (u−, 3u−,−2u−) of system (4.11) are impossible. Moreover, as b increases it is possible loss of stability of the uniform steady state. For a fixed k this occurs when (b− dl2) ( a2 σ2 + l2 )( 1 σ2 + l2 ) + 1 2σ2 ( k − 2b+ √ k2 − 4kb ) × ( − a2 σ2 + (2− 3a2)l2 ) = 0, which is equivalent to Bb2 + Cb+D = 0, EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 17 where B = (2a2 σ4 + (4a2 − 1)l2 σ2 + l4 )2 , C = 2 (2a2 σ4 + (4a2 − 1)l2 σ2 + l4 )( − 2dl2a2 + a2k 2σ4 + −2dl2(a2l2 + l2) + kl2(2− 3a2) 2σ2 − dl6 ) + k σ4 ( − a2 σ2 + (2− 3a2)l2 )2 , D = ( − 2dl2a2 + a2k 2σ4 + −2dl2(a2l2 + l2) + kl2(2− 3a2) 2σ2 − dl6 )2 − k2 4σ4 (− a 2 σ2 + (2− 3a2)l2)2. Then b = bc := −C − √ C2 − 4BD 2B , or b = bc := −C + √ C2 − 4BD 2B . (4.15) Next, we consider the relation between b and l. For convenience, from the expression of equation (4.15) we only consider b = −C+ √ C2−4BD 2B . We can easily to know that b has a minimum, that is, there is a lc, such that bmin = b(lc). In order to see the relation of l and b more clearly, we take the parameter d, k, σ with specific value, and obtain the figure about b and l (see Figure 3). 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 l b b−−−l Figure 3. The phase show the relation between b and l at the different value of σ, the parameter value is d = 1, k = 5 and the green, blue and red curve respectively represents σ = 1.2, 1, 0.8. Thus as b increased beyond bmin, the uniform steady state (u, v, w) = (k + √ k2 − 4kb 2k , 3 k + √ k2 − 4kb 2k , −2 k + √ k2 − 4kb 2k ) looses stability and it is anticipated that a new, non-uniform steady state will appear having a spatial structure similar to exp(ilx). This prove (ii) of Theorem 1.3. � 18 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 Similar to the above process, we can also get the relation between σ and l (we omit the process). Following we study the influence of σ for the solution of the equation (4.1) by using the numerical simulation. As for (K1), we first develop the initial value problem. Set u(x, 0) = { u−, for x ≤ L0, 0, for x > L0. (4.16) 0 10 20 30 40 0 5 10 15 0 1 2 distance x σ=0.25 time t s p e c ie s u 0 10 20 30 40 0 5 10 15 0 0.5 1 1.5 2 distance x σ=0.5 time t s p e c ie s u 0 10 20 30 40 0 5 10 15 0 1 2 distance x σ=1 time t s p e c ie s u 0 20 40 0 5 10 15 0 0.5 1 1.5 2 distance x σ=1.25 time t s p e c ie s u Figure 4. Numerical simulations of the time evolution and space evolution for the bistable nonlocal equation (4.1) with kernel φσ(x) = A σ e − aσ |x| − 1 σ e − |x|σ . The computational domain is x ∈ [0, 40], t ∈ [0, 15]. The parameter values: d = 1, k = 5, b = 1, σ is followed by 0.25, 0.5, 1, 1.25. From the definition of v(x, t), we have v(x, 0) = ∫ R A σ e− a|x−y| σ u(y, 0)dy. Then v(x, 0) = { 3u− − 3u− 2 e a(x−L0) σ , for x ≤ L0, 3u− 2 e− a(x−L0) σ , for x > L0. (4.17) Similarly, we know that w(x, 0) = ∫ R 1 σ e− |x−y| σ u(y, 0)dy . EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 19 Then w(x, 0) = { 2u− − u−e x−L0 σ , for x ≤ L0, u−e − x−L0 σ , for x > L0. (4.18) With (4.16)–(4.18) and the zero-flux boundary conditions, simulating results for (4.11) are also performed by pdepe in Matlab; see Figure 4. From the previous analysis, we know that the state u = u− of the equation (4.1) may be unstable. So what steady state will occur around u = u−? From the Figure 4, we can see that equation (4.1) will have a periodic steady state around u = u−; that is to say, (4.1) admit a traveling wave solution connecting 0 to a periodic steady state. 5. Periodic stationary solutions In Section 4, we showed that the wave can connect 0 to a period steady state through numerical simulations. In this section, we prove that equation (1.1) indeed admits stationary periodic solutions for σ = σc, k = kc and b = bc. Firstly, linearizing (1.1) around u = u−, that is to say, let u = u− + ṽ, we have ṽt = ṽxx + bṽ − ku2 −φ ∗ ṽ − ku0u−ṽφ ∗ ṽ − kṽ2φ ∗ ṽ. Up to a rescaling, let ṽ(t, x) = v(t, σx), we can obtain the new equation vt = σ2vxx + bv − ku2 −φσ ∗ v − ku0u−vφσ ∗ v − kv2φσ ∗ v, where φσ(x) = 1 σφ( xσ ) and v is 2π-periodic in x. We define B(σ, k, b)v := σ2vxx + bv − ku2 −φσ ∗ v, Q(v, k, σ) := −ku0u−vφσ ∗ v − kv2φσ ∗ v. We obtain vt = B(σ, k, b)v + Q(v, k, σ). (5.1) If we define Y := L2 per[0, 2π] = {u ∈ L2 loc(R)|u(x+ 2π) = u(x), x ∈ R}, Υ := D(A) = H2 per[0, 2π] = {u ∈ H2 loc(R)|u(x+ 2π) = u(x), x ∈ R} then we know that Q : Υ→ Υ is smooth. Equation (5.1) can be written as vt = Bcv + C(ε, δ)v + Q(v, kc + ε2, σc + δ), where Bc = B(σc, kc, bc) and C(ε, δ) = B(σc + δ, kc + ε2, bc + ε2 2 ) −B(σc, kc, bc). Since Bc is continuous and Υ is dense and compactly embedded into Y, that the resolvent of Bc is compact and its spectrum σ(Bc) only have eigenvalues λ. From (1.6), we know that σ(Bc) = {λn ∈ C|λn = −σ2n2 + b− ku2 −φ̂(nσ), n ∈ Z}. (5.2) Consequently, σ(Bc) ∩ iR = {0}, and λ = 0 which geometric multiplicity is two, and the corresponding eigenvectors are e(x) := eix and e(x) := e−ix. In addition, the algebraic multiplicity is also two by computation. We define Yc := {e, e} and the spectral projection Hc : Y→ Yc as Hcu = 〈u, e〉e + 〈u, e〉e, 20 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 where 〈u, v〉 = 1 2π ∫ 2π 0 u(x)v(x)dx. It follows from (5.2) that ‖(iν −Bc) −1‖(id−Hc)X ≤ C 1 + |ν| , ν ∈ R, where C > 0 is a positive constant. So, by using the center manifold theorem (see [12, 14, 21]), we know that there exist U ⊂ Yc, V ⊂ (id−Hc)Υ, W ⊂ R2, for any m <∞, a Cm−map Φ : U ×W → V having the following properties. (i) v(t) = B(t)e + B(t)e + Φ(B(t),B(t), ε, δ), t ∈ R. (ii) ‖Φ(B,B, ε, δ)‖Υ = O(|ε|2|B|+ |δ||B|+ |B|2). (iii) dB dt = g(B,B, ε, δ) = Bh(|B|2, ε, δ), where h is a Cm−1 and is a real-valued. From the above three properties, we know that if we want to obtain the periodic steady solution of (5.1), only need to obtain the form of B. Furthermore, if we want to obtain the form of B, only need to obtain the form of h. So we next to find the form of h. Lemma 5.1. The Taylor expansion of the map h is h(|B|2, ε, δ) = ( − kc − 2bc + √ k2 c − 4kcbc 4 φ̂′′(kc)− 1 ) δ2 + ε2 2 + (4bc + ε2)φ̂(σc + δ)√ k2 c − 4kcbc + √ k2 c − 4kcbc − 4bcε2 − ε4 ε2 +$|B|2 +O(|δ|3 + |ε||δ|+ |B|4), (5.3) where $ := −4k2 c + 4kc √ k2 c − 4kcbc −(kc − 4bc)− √ k2 c − 4kcbc + 11k2 c − 124kcbc + 11kc √ k2 c − 4kcbc −(kc − 4bc)− √ k2 c − 4kcbc φ̂(σ) + 2kcbcφ̂(2σ) + 2kc(kc − 2bc + √ k2 c − 4kcbc)(φ̂ 2(σ) + φ̂(σ)φ̂(2σ)) −σ2 + bc − kc−2bc+ √ k2c−4kcbc 2 φ̂(2σ) + kc(kc − 2bc + √ k2 c − 4kcbc) + 6kcbcφ̂(σ) −σ2 + bc − kc−2bc+ √ k2c−4kcbc 2 φ̂(2σ) + 8kc(kc − 2bc + √ k2 c − 4kcbc) −(kc − 4bc)− √ k2 c − 4kcbc φ̂2(σ) < 0. Proof. substituting v(t) = B(t)e+B(t)e+Φ(B(t),B(t), ε, δ) in (5.1) and comparing the coefficient about Be, one obtains dB dt = ( − (σc + δ)2 + bc + ε2 2 − kc − 2bc + √ k2 c − 4kcbc − 4bcε 2 − ε4 2 φ̂(σc + δ) ) B +O(ε,δ)(B|B|2). From the assumption of d(σ, k, b), we know that −σ2 c + bc − kc − 2bc + √ k2 c − 4kcbc 2 φ̂(σc) = 0, −2σc − kc − 2bc + √ k2 c − 4kcbc 2 φ̂′(σc) = 0. EJDE-2020/84 SPATIAL DYNAMICS OF A NONLOCAL BRDE 21 Therefore, ζ(ε, δ) :=− (σc + δ)2 + bc + ε2 2 − kc − 2bc + √ k2 c − 4kcbc − 4bcε2 − ε4 2 φ̂(σc + δ) = ( − kc − 2bc + √ k2 c − 4kcbc 4 φ̂′′(kc)− 1 ) δ2 + ε2 2 + (4bc + ε2)φ̂(σc + δ)√ k2 c − 4kcbc + √ k2 c − 4kcbc − 4bcε2 − ε4 ε2 +O(|ε|2|δ|+ |δ|3), as (ε, δ)→ (0, 0). To obtain $ in (5.3), let (ε, δ) = (0, 0). Note that v(t) = B(t)e + B(t)e + B2(t)e2,0 + BBe1,1 + B 2 e0,2 +O(|B|3). and $ is actually the coefficient of the term B|B ·B|. From (5.1), it follows that this term appears in −kcu0v 2, −2kcu−vφ ∗ v and −kcv2φ ∗ v, thus $ =〈−2kcu−(e · φ ∗ e1,1 + e · φ ∗ e2,0 + e2,0 · φ ∗ e + e1,1 · φ ∗ e), e〉 − 〈k(e · e · φ ∗ e + 2e · e · φ ∗ e) + kcu0(2e · e1,1 + 2e · e2,0), e〉. (5.4) Next, we need to compute e1,1 and e2,0. Straightforward computations show that e1,1(x) = 2kcu0 + 4kcu−φ̂(σc) bc − kcu2 − + Span(e, e), e2,0(x) = kcu0 + 2kcu−φ̂(σc) −σ2 c + bc − kcu2 −φ̂(2σc) ei2x + Span(e, e). Using e1,1 and e2,0 in (5.4), we obtain the coefficient $. This completes the proof. � Proof of Theorem 1.4. For convenience, we define Λ := ( − kc − 2bc + √ k2 c − 4kcbc 4 φ̂′′(kc)− 1 )δ2 $ + ε2 2$ + (4bc + ε2)φ̂(σc + δ)√ k2 c − 4kcbc + √ k2 c − 4kcbc − 4bcε2 − ε4 ε2 $ > 0. (5.5) We aim at finding a nontrivial stationary solution B0 ∈ C satisfying 0 = h(|B|2, ε, δ). (5.6) Up to a rescaling B0 = √ ΛB̃0, equation (5.6) can be rewritten as Λ · (−$ +$|B̃0|2 +O( √ Λ)) = 0, as Λ→ 0. By using the implicit function theorem, we have |B̃0| = 1 +O( √ Λ), as Λ→ 0. So equation (5.1) admits periodic solutions of the form vε,δ(x) = √ Λ cos((σc + δ)x) +O(|Q|), 22 B.-S. HAN, M.-X. CHANG, Y. YANG EJDE-2020/84 where Q = ( − kc − 2bc + √ k2 c − 4kcbc 4 φ̂′′(kc)− 1 ) δ2 + ε2 2 + (4bc + ε2)φ̂(σc + δ)√ k2 c − 4kcbc + √ k2 c − 4kcbc − 4bcε2 − ε4 ε2, (5.7) for some ε ∈ (0, ε0] and δ satisfying( − kc − 2bc + √ k2 c − 4kcbc 4 φ̂′′(kc)− 1 ) δ2 < (4bc + ε2)φ̂(σc + δ)√ k2 c − 4kcbc + √ k2 c − 4kcbc − 4bcε2 − ε4 ε2 + ε2 2 . So we have the existence of periodic solutions of (1.1) that can be written as uε,δ(x) = u− + √ Λ cos((σc + δ)x) +O(|Q|). This completes the proof. � Acknowledgements. This work was supported by the Natural Science Founda- tion of China (11801470, 31700347), by the Fundamental Research Funds for the Central Universities (2682018CX64), and by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190578). References [1] S. 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Differential Equations, 222 (2006), no. 1, 185–232. [30] S.-L. Wu, C.-H. Hsu; Entire solutions with annihilating fronts to a nonlocal dispersal equation with bistable nonlinearity and spatio-temporal delay, J. Dynam. Differential Equations, 29 (2017), no. 2, 409–430. [31] C. Zhang, B. Han; Stability analysis of stochastic delayed complex networks with multi- weights based on Razumikhin technique and graph update theory, Physica A, 538 (2020), 122827. Bang-Sheng Han (corresponding author) School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, China Email address: hanbangsheng@swjtu.edu.cn Meng-Xue Chang School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, China Email address: mengxue chang@163.com Yinghui Yang School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, China Email address: yangyh8605@swjtu.edu.cn 1. Introduction 2. Existence of traveling wave solutions 2.1. A priori estimates of solution u in a finite domain 2.2. Construction of a solution (c,u) 3. Behavior of the solution at 4. Linear stability analysis and numerical simulations 5. Periodic stationary solutions Acknowledgements References