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Research article

Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity

  • Received: 20 September 2020 Accepted: 27 October 2020 Published: 03 November 2020
  • MSC : 35J20, 35J65, 35R11

  • In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity {(a+bΩ2|u(x)u(y)|2|xy|N+2sdxdy)(Δ)su+V(x)u=Q(x)|u|p2ulnu2,in Ω,u=0,in RNΩ, where ΩRN is a smooth bounded domain, N>2s (0<s<1), (Δ)s is the fractional Laplacian, V,Q are continuous, V,Q0. a,b>0 are constants, 4<p<2s:=2NN2s. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.

    Citation: Qing Yang, Chuanzhi Bai. Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity[J]. AIMS Mathematics, 2021, 6(1): 868-881. doi: 10.3934/math.2021051

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  • In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity {(a+bΩ2|u(x)u(y)|2|xy|N+2sdxdy)(Δ)su+V(x)u=Q(x)|u|p2ulnu2,in Ω,u=0,in RNΩ, where ΩRN is a smooth bounded domain, N>2s (0<s<1), (Δ)s is the fractional Laplacian, V,Q are continuous, V,Q0. a,b>0 are constants, 4<p<2s:=2NN2s. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.


    Popular social networks play an essential role in our daily lives. In recent years, many rumors are spreading on social networks such that our lives are seriously affected. In order to control rumor propagation in social networks, we should understand the process of information propagation. Hence, some mathematical models were proposed to characterize and predict the process of information propagation in online social networks, such as, [26,27,17]. In [17], Wang et al. proposed the following diffusive logistic model:

    {ut=duxx+r(t)u(1u/K),t>1, l<x<L,u(1,x)=u0(x),lxL,ux(t,l)=0, ux(t,L)=0,t>1,

    where r, K and d represent the intrinsic growth rate, the carrying capacity, and the diffusion rate, respectively. l and L stand for the upper and lower bounds of the distances between the source s and other social networks users.

    In above system, l and L are fixed boundary and so information only spreads in this fixed area. But in reality, the spreading area of information is changing with time. This can be addressed by considering this over the varying domain. In 2013, Lei et al. [7] introduced the free boundary to study single information diffusion in online social networks,

    {ut=duxx+r(t)u(1u/K),t>0, 0<x<h(t),ux(t,0)=0, u(t,h(t))=0,t>0,h(t)=μux(t,h(t)),t>0,h(0)=h0, u(0,x)=u0(x),0xh0. (1)

    They presented some sharp criteria for information spreading and vanishing. Furthermore, if the information spreading happens, they gave the asymptotic spreading speed which is determined by a corresponding elliptic equation.

    The deduction of free boundary condition in (1) can be found in [2]. In 2010, this condition was introduced by Du and Lin [5] to describe the spreading of the invasive species, and a spreading-vanishing dichotomy was first established. After the work of [5] for a logistic type local diffusion model, free boundary approaches to local diffusion problems similar to problem (1) have been studied by many researchers recently. Among the many further extensions, we only mention the extension to certain Lotka-Volterra two-species systems [20,21,22,23] and the references therein.

    The works of [17] and [7] all discussed the spreading of the single information. However, in many practical situations, considering multiple information diffusion process in online social networks is more realistic. In 2013, Peng et al. [14] studied information diffusion initiated from multiple sources in online social networks by numerical simulation. But there are many challenging problems in modeling and analysing multiple information diffusion process. In particular, a simple case was considered by Ren et al. [15]. They assumed that there are three pieces of information A, B and C sent from different sources to compete for influence on online users, where the official information C is viewed as an intervention from the media or government to control the spread of the ordinary information A and B. For simplicity, they further assumed that A and B has no influence on C, A and B compete for influence on each other. Following the approach of [7], they proposed the following model

    {ut=d1uxx+u(a1b1uc1vr1w),t>0, 0<x<h(t),vt=d2vxx+v(a2b2vc2ur2w),t>0, 0<x<h(t),wt=d3wxx+w(a3b3w),t>0, 0<x<h(t),ux(t,0)=vx(t,0)=wx(t,0)=0,t0,u(t,h(t))=v(t,h(t))=w(t,h(t))=0,t0,h(t)=μ[ρ1ux(t,h(t))+ρ2vx(t,h(t))+wx(t,h(t))],t>0,h(0)=h0,u(0,x)=u0(x), v(0,x)=v0(x), w(0,x)=w0(x),0<x<h0, (2)

    where u(t,x),v(t,x),w(t,x) represent the density of influenced users of information A, B, C at time t and location x respectively, h(t) is the spreading front of the news, di (i=1,2,3) is the diffusion rates, ai (i=1,2,3) is the intrinsic growth rates, 1/bi (i=1,2,3) is the carrying capacities, ci (i=1,2) and ri (i=1,2) are the intervention rates, μ stands for the expanding capacity of information. In [15], they first gave the long time behavior of the information: all information spread; one ordinary information and official information spread, while the other ordinary information vanishes; two pieces of ordinary information vanish and official information spreads. And then they established the criteria for spreading and vanishing. Furthermore, they provided some estimates of asymptotic spreading speed when spreading happens. Finally, by some numerical simulations, they illustrated the results and all cases of the asymptotic behavior of the solution.

    Note that in (2), the dispersal of the information is assumed to follow the rules of random diffusion, which is not realistic in general. This kind of dispersal may be better described by a nonlocal diffusion operator of the form

    dRJ(xy)u(t,y)dydu(t,x),

    which can capture short-range as well as long-range factors in the dispersal by choosing the kernel function J properly [1,10,11,12,13,16,24,25].

    Recently, Cao et al. [3] proposed a nonlocal version of the logistic model of [5], and successfully extended many basic results of [5] to the nonlocal model. Motivated by the work [3], some related models with nonlocal diffusion and free boundaries have been considered in several recent works (see, for example, [6,8,9,18,19]). In this paper, following the approach of [3], we propose and examine a nonlocal version of (2), which has the form

    {ut=d1h(t)g(t)J1(xy)u(t,y)dyd1u+u(a1b1uc1vr1w),t>0, g(t)<x<h(t),vt=d2h(t)g(t)J2(xy)v(t,y)dyd2v+v(a2b2vc2ur2w),t>0, g(t)<x<h(t),wt=d3h(t)g(t)J3(xy)w(t,y)dyd3w+w(a3b3w),t>0, g(t)<x<h(t),u(t,x)=v(t,x)=w(t,x)=0,t0, x=g(t) or h(t),g(t)=μh(t)g(t)g(t)[ρ1J1(xy)u(t,x)+ρ2J2(xy)v(t,x)+J3(xy)w(t,x)]dydx,t>0,h(t)=μh(t)g(t)+h(t)[ρ1J1(xy)u(t,x)+ρ2J2(xy)v(t,x)+J3(xy)w(t,x)]dydx,t>0,g(0)=h(0)=h0,u(0,x)=u0(x),v(0,x)=v0(x),w(0,x)=w0(x),h0<x<h0, (3)

    where di (i=1,2,3), ai (i=1,2,3), bi (i=1,2,3), ci (i=1,2), ri (i=1,2), ρi (i=1,2), μ and h0 are given positive constants. The initial functions u0(x), v0(x) and w0(x) belong to

    X(h0):={u0C([h0,h0]) : u0(±h0)=0, u0>0 in (h0,h0)},

    where [h0,h0] represents the initial range of the information. Assumed that u,v,w are identically 0 for xR[g(t),h(t)], and the kernel function Ji:RR (i=1,2,3) is continuous and nonnegative, and have the properties

    (J) J(0)>0, RJ(x)dx=1, J is symmetric, supRJ<.

    The main results of this paper are the following theorems:

    Theorem 1.1 (Global existence and uniqueness). Suppose that Ji(i=1,2,3) satisfies (J). Then for any given h0>0 and u0(x),v0(x),w0(x) belonging to X(h0), problem (3) admits a unique solution (u(t,x),v(t,x),w(t,x),g(t),h(t)) defined for all t>0.

    Theorem 1.2 (Spreading-vanishing dichotomy). Let the conditions of Theorem 1.1 hold and (u,v,w,g,h) be the unique solution of (3). Assume further that J1(x)>0, J2(x)>0 in R, then one of the following alternatives must happen:

    (i) Spreading: limt[h(t)g(t)]=.

    (ii) Vanishing: limt(g(t),h(t))=(g,h) is a finite interval,

    limtmaxg(t)xh(t)u(t,x)=0, limtmaxg(t)xh(t)v(t,x)=0

    and

    limtmaxg(t)xh(t)w(t,x)=0.

    Theorem 1.3 (Spreading-vanishing criteria). Assume that Ji (i=1,2,3) satisfies (J), and J1(x)>0, J2(x)>0 in R. Then the dichotomy in Theorem 1.2 can be determined as follows:

    (i) If a1d1 or a2d2 or a3d3, then necessarily hg=.

    (ii) If ai<difor  i=1,2,3, then

    (a) If hg<, then hgl.

    (b) If h0l/2, then hg=.

    (c) If h0<l/2, then there exist two positive numbers μμ>0 such that hg< when 0<μμ and μ=μ, and hg= when μ>μ,

    where l is given by (13).

    Theorem 1.4 (Asymptotic limit). Let (u,v,w,g,h) be the unique solution of (3) and suppose limt[h(t)g(t)]=. The following conclusions hold:

    (i) If a1>c1a2b2+r1a3b3 and a2>c2a1b1+r2a3b3, then

    limt(u(t,x),v(t,x),w(t,x))= (b2(a1b3r1a3)c1(a2b3r2a3)b3(b1b2c1c2),c2(a1b3r1a3)+b1(a2b3r2a3)b3(b1b2c1c2),a3b3)

    locally uniformly for xR.

    (ii) If a1+c1b2(c2a1b1+r2a3b3)c1a2b2+r1a3b3 and a2>c2a1b1+r2a3b3, then

    limt(u(t,x),v(t,x),w(t,x))=(0,a2b3r2a3b2b3,a3b3) locally uniformly for xR.

    (iii) If a1>c1a2b2+r1a3b3 and a2+c2b1(c1a2b2+r1a3b3)c2a1b1+r2a3b3, then

    limt(u(t,x),v(t,x),w(t,x))=(a1b3r1a3b1b3,0,a3b3) locally uniformly for xR.

    (iv) If a1r1a3b3 and a2r2a3b3, then

    limt(u(t,x),v(t,x),w(t,x))=(0,0,a3b3) locally uniformly for xR.

    Remark 1. Note that for the corresponding local diffusion model in [15], no matter how small the diffusion coefficient di is, vanishing can always happen if h0 and μ are both sufficiently small. However, for (3), Theorem 1.3 indicates that when d1a1 or d2a2 or d3a3, spreading always happens no mater how small h0 and μ are. This is different from the local diffusion model in [15].

    The rest of this paper is organised as follows. In Section 2 we prove Theorem 1.1, namely, problem (3) has a unique solution defined for all t>0. The long-time dynamical behaviour of (3) is investigated in Section 3, where Theorems 1.2, 1.3 and 1.4 are proved. Finally, we conclude this paper with a brief discussion in Section 4.

    For convenience, we first introduce some notations. For given T>0, define

    HT:={hC1([0,T]):h(0)=h0,h(t) is strictly increasing},GT:={gC1([0,T]):gHT},DT=Dg,hT:={(t,x)R2:0<t<T, g(t)<x<h(t)}.

    The proof of Theorem 1.1. The existence and uniqueness of solution to the problem (3) can be done in a similar fashion as in [3,6]. We only list the main steps in the proof.

    Noting that J3 satisfies (J), f3(w):=w(a3b3w) satisfies (f1) and (f2) in [3], and w0(x) belongs to X(h0) for any h0>0. For any given T>0 and (g,h)GT×HT, it follows from [3,Lemma 2.3] that the following problem

    {wt=d3h(t)g(t)J3(xy)w(t,y)dyd3w+w(a3b3w),0<t<T,g(t)<x<h(t),w(t,g(t))=w(t,h(t))=0,0<t<T,w(0,x)=w0(x),h0xh0

    admits a unique solution w(t,x), and

    0<w(t,x)max{w0,a3/b3}=:A3 in DT.

    For such w(t,x), it is easy to check that f1(t,x,u,v):=u(a1b1uc1v+r1w(t,x)) and f2(t,x,u,v):=v(a2b2vc2u+r2w(t,x)) satisfy (f), (f1) and (f2) in [6]. For (g,h) given above, it follows from [6,Lemma 2.3] that the following problem

    {ut=d1h(t)g(t)J1(xy)u(t,y)dyd1u+u(a1b1uc1vr1w(t,x)),0<t<T, g(t)<x<h(t),vt=d2h(t)g(t)J2(xy)v(t,y)dyd2v+v(a2b2vc2ur2w(t,x)),0<t<T, g(t)<x<h(t),u(t,x)=v(t,x)=0,0<t<T, x=g(t) or h(t),u(0,x)=u0(x), v(0,x)=v0(x),h0xh0

    has a unique solution (u,v) and

    0<umax{u0,a1/b1}=:A1 in DT,
    0<vmax{v0,a2/b2}=:A2 in DT.

    For (u,v,w,g,h) above, we define (˜g,˜h) for t[0,T] by

    {˜g(t):=h0μt0h(τ)g(τ)g(τ)[ρ1J1(xy)u(τ,x)+ρ2J2(xy)v(τ,x)+J3(xy)w(τ,x)]dydxdτ,˜h(t):=h0+μt0h(τ)g(τ)+h(τ)[ρ1J1(xy)u(τ,x)+ρ2J2(xy)v(τ,x)+J3(xy)w(τ,x)]dydxdτ.

    Since Ji (i=1,2,3) satisfies (J), there exist constants ϵ0(0,h0/4) and δ0 such that

    Ji(x)δ0 if |x|ϵ0, i=1,2,3.

    Let

    L:=(b1+c2)A1+(c1+b2)A2+(r1+r2+b3)A3,

    then

    f1(u,v,w)=u(a1b1uc1vr1w)(b1A1+c1A2+r1A3)uLu,f2(u,v,w)=v(a2b2vc2ur2w)(b2A2+c2A1+r2A3)vLv,f3(w)=w(a3b3w)b3A3wLw.

    Using this we can follow the corresponding arguments of [6] to show that, for some sufficiently small T0=T0(μ,A1,A2,A3,h0,ϵ0,ρ1,ρ2,J)>0 and any T(0,T0],

    sup0t1<t2T˜g(t2)˜g(t1)t2t1˜σ0,   inf0t1<t2T˜h(t2)˜h(t1)t2t1σ0,
    ˜h(t)˜g(t)2h0+ϵ04 for t[0,T],

    where

    ˜σ0=14ϵ0δ0μe(d1+d2+d3+L)T0h0+ϵ04h0 [ρ1u0(x)+ρ2v0(x)+w0(x)]dx,
    σ0=14ϵ0δ0μe(d1+d2+d3+L)T0h0h0ϵ04[ρ1u0(x)+ρ2v0(x)+w0(x)]dx.

    Let

    ΣT:={(g,h)GT×HT : sup0t1<t2Tg(t2)g(t1)t2t1˜σ0,inf0t1<t2Th(t2)h(t1)t2t1σ0, h(t)g(t)2h0+ϵ04 for t[0,T]},

    and define the mapping

    F(g,h)=(˜g,˜h).

    Then the above analysis indicates that

    F(ΣT)ΣT for T(0,T0].

    Next, we will show that F is a contraction mapping on ΣT for sufficiently small T(0,T0]. For any given (gi,hi)ΣT (i=1,2), denote

    (˜gi,˜hi)=F(gi,hi).

    Let

    U(t,x):=u1(t,x)u2(t,x), V(t,x):=v1(t,x)v2(t,x),W(t,x):=w1(t,x)w2(t,x).

    Then we can follow the approach of Step 2 in the proof of [3,Theorem 2.1] to show

    |˜g1(t)˜g2(t)|+|˜h1(t)˜h2(t)| 6h0μρ1TUC([0,T]×R)+6h0μρ2TVC([0,T]×R)+6h0μTWC([0,T]×R)+3Tμ(ρ1A1+ρ2A2+A3)[g1g2C([0,T])+h1h2C([0,T])].

    We can apply the same argument as the step 2 in the proof of [6,Theorem 2.1] to obtain that there exist C1 and T1(0,T0) such that, for TT1,

    max{UC([0,T]×R), VC([0,T]×R)}C1[g1g2C([0,T])+h1h2C([0,T])].

    Meanwhile, by using the same argument as the step 2 in the proof of [3,Theorem 2.1] to obtain that there exist C2 and T2(0,T1) such that, for TT2,

    WC([0,T]×R)C2[g1g2C([0,T])+h1h2C([0,T])].

    Then we have, for TT2,

    |˜g1(t)˜g2(t)|+|˜h1(t)˜h2(t)| C3T[g1g2C([0,T])+h1h2C([0,T])],

    where

    C3=6h0μρ1C1+6h0μρ2C1+6h0μC2+3μ(ρ1A1+ρ2A2+A3).

    This shows that if we choose ˜T such that

    0<˜Tmin{T2,12C3},

    then, for T(0,˜T],

    |˜g1(t)˜g2(t)|+|˜h1(t)˜h2(t)| 12[g1g2C([0,T])+h1h2C([0,T])]

    and so F is a contraction mapping on ΣT. Hence F has a unique fixed point (g,h) in ΣT, which gives a nonnegative solution (u,v,w,g,h) of (3) for t(0,T]. Similar to Steps 3 and 4 in the proof of [6,Theorem 2.1], we can show that this is the unique solution of (3) and it can be extended uniquely to all t>0.

    Since u,v and w are positive in DT, we have h(t)>0 and g(t)<0 for t>0. Thus we can define

    limtg(t)=g[,h0),   limth(t)=h(h0,].

    Clearly we have either

    (i) hg<, or (ii) hg=.

    We will call (ⅰ) the vanishing case, and call (ⅱ) the spreading case. The main purpose of this section is to determine when (ⅰ) or (ⅱ) can occur, and to determine the long-time profile of (u,v,w) if (ⅰ) or (ⅱ) happens.

    Before analysing the vanishing phenomenon, we first give some lemmas.

    Lemma 3.1. Let the condition (J) hold for the kernel functions Ji (i=1,2,3), and β1,β2,β3>0 be constants. Suppose that g,hC1([0,)), g(0)<h(0), g(t)0, h(t)0 and wi,witC(D)L(D) for i=1,2,3, where D={t>0, g(t)<x<h(t)}. If (w1,w2,w3,g,h) satisfies

    {h(t)=3i=1βih(t)g(t)h(t)Ji(xy)wi(t,x)dydx,t0,g(t)=3i=1βih(t)g(t)g(t)Ji(xy)wi(t,x)dydx,t0, (4)

    and

    limth(t)limtg(t)<, (5)

    then

    limtg(t)=limth(t)=0.

    This lemma can be proven by using the same arguments in [6,Lemma 3.1]. Next we recall another lemma which will be used later.

    Lemma 3.2. ([6,Lemma 3.2]) Let J satisfy the condition (J) and J(x)>0 in R. Suppose that g,hC1([0,)), g(0)<h(0), g(t)0, h(t)0, and (5) holds. If (w,g,h) satisfies, for some positive constants β and M,

    0wM in  D,   w(t,g(t))=w(t,h(t))=0,   t0,
    h(t)βh(t)g(t)h(t)J(xy)w(t,x)dydx,   t>0,

    and limth(t)=0, then

    limth(t)g(t)w(t,x)dx=0,   0h(t)g(t)w(t,x)dxdt<.

    We define the operator LdiΩ+β:C(¯Ω)C(¯Ω) by

    (LdiΩ+β)[ϕ](x):=diΩJi(xy)ϕ(y)dydiϕ(x)+β(x)ϕ(x),

    where Ω is an open bounded interval in R, and βC(¯Ω). The generalized principal eigenvalue of LdiΩ+β is given by

    λp(LdiΩ+β):=inf{λR:(LdiΩ+β)[ϕ]λϕ in Ω for some ϕC(¯Ω), ϕ>0}.

    Then we will use the techniques in [6,Theorem 3.3] to give the vanishing result.

    Lemma 3.3. Assume that Ji (i=1,2,3) satisfies (J), Ji(x)>0 (i=1,2) in R. Let (u,v,w,g,h) be the unique solution of (3). If hg<, then

    limtmaxg(t)xh(t)u(t,x)=limtmaxg(t)xh(t)v(t,x)=limtmaxg(t)xh(t)w(t,x)=0, (6)

    moreover,

    λp(Ldi(g,h)+ai)0, i=1,2,3. (7)

    Proof. By the similar arguments in [3,Theorem 3.7], we can have

    limtmaxg(t)xh(t)w(t,x)=0 and λp(Ld3(g,h)+a3)0.

    In the following, we only prove

    limtmaxg(t)xh(t)u(t,x)=0 and λp(Ld1(g,h)+a1)0. (8)

    The conclusion for v can be obtained similarly, so we omit here.

    By the same arguments in [6], we can have

    \begin{equation} \lim\limits_{t\rightarrow\infty}u(t, x) = \lim\limits_{t\rightarrow\infty}v(t, x) = \lim\limits_{t\rightarrow\infty}w(t, x) = 0 \ \text{for almost every}\ x\in[g(0), h(0)]. \end{equation} (9)

    Define

    M(t): = \max\limits_{x\in[g(t), h(t)]}u(t, x)

    and

    X(t): = \{x\in(g(t), h(t)): u(t, x) = M(t)\}.

    Then X_i(t) is a compact set for each t>0 . Therefore, there exist \underline\xi_i(t), \;\overline\xi_i(t)\in X_i(t) such that

    u_t(t, \overline{\xi}(t)) = \max\limits_{x\in X(t)}u_t(t, x), \ u_t(t, \underline{\xi}(t)) = \min\limits_{x\in X(t)}u_t(t, x).

    By the arguments in [6], the following claim holds

    \begin{align*} &M'(t+0): = \lim\limits_{s > t, s\rightarrow t}\frac{M(s)-M(t)}{s-t} = u_t(t, \overline{\xi}(t)), \\ &M'(t-0): = \lim\limits_{s < t, s\rightarrow t}\frac{M(s)-M(t)}{s-t} = u_t(t, \underline{\xi}(t)). \end{align*}

    If M(t) has a local maximum at t = t_0 , then M'(t_0) exists and M'(t_0) = 0 . Moreover, if M(t) is monotone nondecreasing for all large t and \lim\limits_{t\rightarrow\infty}M(t) = \sigma>0 , then M'(t-0)\rightarrow0 as t\rightarrow\infty ; if M(t) is monotone nonincreasing for all large t and \lim\limits_{t\rightarrow\infty}M(t) = \sigma>0 , then M'(t+0)\rightarrow0 as t\rightarrow\infty .

    Now we are ready to show that \lim\limits_{t\rightarrow\infty}M(t) = 0 . This can be done by the similar argument in Theorem 3.3 of [6]. Arguing indirectly we assume that this claim does not hold. Then

    \begin{equation} \sigma^\ast: = \limsup\limits_{t\rightarrow\infty}M(t)\in(0, \infty). \end{equation} (10)

    By the above stated properties of M(t) , there exists a sequence t_n>0 increasing to \infty as n\rightarrow\infty , and \xi_n\in\{\overline{\xi}(t_n), \underline{\xi}(t_n)\} such that

    \lim\limits_{n\rightarrow\infty}u(t_n, \xi_n) = \sigma^\ast, \ \lim\limits_{n\rightarrow\infty}u_t(t_n, \xi_n) = 0.

    By passing to a subsequence of (t_n, \xi_n) if necessary, we may assume, without loss of generality,

    \lim\limits_{n\to\infty}v(t_n, \xi_n) = \rho\in[0, \infty).

    By Lemma 3.1, we have \lim\limits_{t\rightarrow\infty}h'(t) = 0 . It follows from this fact and Lemma 3.2 that

    \lim\limits_{t\rightarrow\infty} \int_{g(t)}^{h(t)}u(t, y)dy = 0.

    Since \sup\limits_{x\in \mathbb R}J_1(x)<\infty by \textbf{(J)} , we have

    \lim\limits_{t\rightarrow\infty} \int_{g(t)}^{h(t)}J_1(x-y)u(t, y)dy = 0 \ \text{uniformly for}\ x\in \mathbb R.

    We now make use of the identity

    u_t = d_1 \int_{g(t)}^{h(t)}J_1(x-y)u(t, y)dy-d_1u +u(a_1-b_1u-c_1v-r_1w)

    with (t, x) = (t_n, \xi_n) . Letting n\rightarrow\infty , we obtain

    0\leq-d_1\sigma^\ast+\sigma^\ast(a_1-b_1\sigma^\ast-c_1\rho) < \sigma^\ast(a_1-d_1).

    It follows that a_1>d_1 . We show next that this leads to a contradiction.

    Indeed, by (9), there exists x_0\in (g(0), h(0)) such that

    \lim\limits_{t\to\infty}u(t, x_0) = \lim\limits_{t\to\infty}v(t, x_0) = \lim\limits_{t\to\infty}w(t, x_0) = 0.

    Therefore we can find T>0 large so that

    -d_1+a_1-b_1u(t, x_0)-c_1v(t, x_0)-r_1w(t, x_0) > (a_1-d_1)/2 > 0 \ \text{for}\ t\geq T.

    It then follows from the equation satisfied by u that

    u_t(t, x_0)\geq \frac{a_1-d_1}{2}u(t, x_0) \ \mbox{ for } t\geq T,

    which implies u(t, x_0)\to\infty as t\to\infty , a contradiction to the boundedness of u . This completes the proof of \lim\limits_{t\to\infty}\max\limits_{g(t)\leq x\leq h(t)}\!u(t, x) = 0 . Similarly, \lim\limits_{t\to\infty}\max\limits_{g(t)\leq x\leq h(t)}\!v(t, x) = 0 .

    In the following we prove the second conclusion of (8). Suppose on the contrary that \lambda_p(\mathcal{L}_{(g_{\infty}, h_{\infty})}^{d_1}+a_1)>0 . Then there exists small \epsilon_1\in(0, \frac{2a_1}{c_1+r_1}) such that

    \lambda_p(\mathcal{L}_{(g_{\infty}+\epsilon, h_{\infty}-\epsilon)}^{d_1} +a_1-\frac{\epsilon}{2}(c_1+r_1)) > 0 \ \text{for}\ \epsilon\in(0, \epsilon_1).

    Moreover, for such \epsilon , it follows from

    h_\infty-g_\infty < \infty \ \text{and}\ \lim\limits_{t\to\infty}v(t, x) = \lim\limits_{t\to\infty}w(t, x) = 0 \ \text{for}\ x\in \mathbb R

    that there exists T_\epsilon such that

    g(t) < g_\infty+\epsilon, \ h(t) > h_\infty-\epsilon \ \text{for}\ t > T_\epsilon,

    and

    v(t, x)\leq\frac{\epsilon}{2}, \ w(t, x)\leq\frac{\epsilon}{2} \ \text{for}\ t > T_\epsilon \ \text{and}\ x\in \mathbb R.

    Then

    \begin{align*} u_t\geq d_1 \int_{g_{\infty}+\epsilon} ^{h_{\infty}-\epsilon}J_1(x-y)u(t, y)dy-d_1u&\\ +u[a_1-b_1u-\frac{\epsilon}{2}(c_1+r_1)]&, \ \ t > T_\epsilon, \ x\in[g_{\infty}+\epsilon, h_{\infty}-\epsilon]. \end{align*}

    Let \phi(x) be the corresponding normalized eigenfunction of \lambda_p(\mathcal{L}_{(g_{\infty}+\epsilon, h_{\infty}-\epsilon)}^{d_1} +a_1-\frac{\epsilon}{2}(c_1+r_1)) , namely, \|\phi\|_\infty = 1 and

    d_1 \int_{g_{\infty}+\epsilon} ^{h_{\infty}-\epsilon}J_1(x-y)\phi(y)dy-d_1\phi(x) +(a_1-\frac{\epsilon}{2}(c_1+r_1))\phi(x) = \lambda_p\phi(x).

    Then, for any \delta>0 ,

    d_1 \int_{g_{\infty}+\epsilon} ^{h_{\infty}-\epsilon}J_1(x-y)\delta\phi(y)dy-d_1\delta\phi(x) +(a_1-\frac{\epsilon}{2}(c_1+r_1))\delta\phi(x) = \lambda_p\delta\phi(x) > 0.

    If we choose \delta small enough such that \delta\phi(x)\leq u(T_\epsilon, x) for x\in[g_{\infty}+\epsilon, h_{\infty}-\epsilon] , then we can use [3,Lemma 3.3] and a simple comparison argument to obtain

    u(t, x)\geq\delta\phi(x) > 0 \ \text{for}\ t > T_\epsilon \ \text{and}\ x\in[g_{\infty}+\epsilon, h_{\infty}-\epsilon].

    This is a contradiction to \lim\limits_{t\to\infty}\max\limits_{g(t)\leq x\leq h(t)}u(t, x) = 0 . Thus \lambda_p\left(\mathcal{L}^{d_1}_{(g_\infty, h_\infty)}+a_1\right) \leq 0 .

    Then Theorem 1.2 can be obtained by Lemma 3.3 directly.

    Corollary 1. Suppose that J_1, J_2 and J_3 satisfy the conditions in Lemma 3.3, and (u, v, w, g, h) is the unique solution of (3). If a_1\geq d_1 or a_2\geq d_2 or a_3\geq d_3 , then necessarily h_\infty-g_\infty = \infty .

    Proof. Arguing indirectly we assume that h_\infty-g_\infty<\infty and a_i\geq d_i for some i\in\{1, 2, 3\} . Thanks to [3,Proposition 3.4],

    \lambda_p\left(\mathcal{L}^{d_i}_{(g_\infty, h_\infty)}+a_i\right) > 0.

    This is a contradiction to Lemma 3.3.

    Hence, Theorem 1.3 (ⅰ) has been proved.

    We next consider the case that

    \begin{equation} a_i < d_i \ \text{for}\ i = 1, 2, 3. \end{equation} (11)

    In this case, it follows from [3,Proposition 3.4] that there exists l_i\ (i = 1, 2, 3) such that

    \begin{equation} \begin{cases} \lambda_{p}(\mathcal{L}_{(0, l_i)}^{d_i}+a_i) = 0, &\mbox{ if } l = l_i, \\ (l-l_i)\lambda_{p}(\mathcal{L}_{(0, l_i)}^{d_i}+a_i) > 0, & \mbox{ if } l-l_i\in (0, +\infty)\backslash\{l_i\}. \end{cases} \end{equation} (12)

    Define

    \begin{equation} l_* = \min\{l_1, l_2, l_3\}. \end{equation} (13)

    It is easily seen that conclusions (a) and (b) of Theorem 1.3 follow directly from the definition of l_* , (12) and Lemma 3.3. In the following, we prove Theorem 1.3 (c) by several lemmas.

    Lemma 3.4. Under the assumptions of Theorem 1.3, if h_0<l_*/2 , then there exists a positive number \mu_0 such that h_\infty-g_\infty<\infty for any \mu\in(0, \mu_0] .

    We need some comparison results to prove this lemma. The proof of the following Lemma 3.5 can be carried out by the same arguments in the proof of [3,Theorem 3.1]. Since the adaptation is rather straightforward, we omit the details here.

    Lemma 3.5. For T\in(0, +\infty) , suppose that \overline{g}, \overline{h}\in C([0, T]) , \overline{u}, \overline{v}\in C(\overline{D_T^{\overline{g}, \overline{h}}}) . If (\overline{u}, \overline{v}, \overline{w}, \overline{g}, \overline{h}) satisfies

    \begin{equation*} \begin{cases} \overline{u}_{t} \geq d_1 \int_{\overline{g}(t)}^{\overline{h}(t)} J_1(x-y)\overline{u}(t, y)dy-d_1\overline{u} +\overline{u}(a_1-b_1\overline{u}), &t > 0, \ \overline{g}(t) < x < \overline{h}(t), \\ \overline{v}_{t} \geq d_2 \int_{\overline{g}(t)}^{\overline{h}(t)} J_2(x-y)\overline{v}(t, y)dy-d_2\overline{v} +\overline{v}(a_2-b_2\overline{v}), &t > 0, \ \overline{g}(t) < x < \overline{h}(t), \\ \overline{w}_{t} \geq d_3 \int_{\overline{g}(t)}^{\overline{h}(t)} J_3(x-y)\overline{w}(t, y)dy-d_3\overline{w} +\overline{w}(a_3-b_3\overline{w}), &t > 0, \ \overline{g}(t) < x < \overline{h}(t), \\ \overline{u}(t, x)\geq0, \ \overline{v}(t, x)\geq0, \ \overline{w}(t, x)\geq0, &t\geq 0, \ x = \overline{g}(t) \mathit{\text{or}} \ \overline{h}(t), \\ \overline{g}'(t)\leq-\mu \int_{\overline{g}(t)}^{\overline{h}(t)} \int_{-\infty}^{\overline{g}(t)} [\rho_1J_1(x-y)\overline{u}(t, x)&\\ \quad +\rho_2J_2(x-y)\overline{v}(t, x) +J_3(x-y)\overline{w}(t, x)]dydx, &t > 0, \\ \overline{h}'(t)\geq\mu \int_{\overline{g}(t)}^{\overline{h}(t)} \int_{\overline{h}(t)}^{+\infty} [\rho_1J_1(x-y)\overline{u}(t, x)&\\ \quad +\rho_2J_2(x-y)\overline{v}(t, x) +J_3(x-y)\overline{w}(t, x)]dydx, &t > 0, \\ \overline{g}(0)\leq -h_0, \ \overline{h}(0)\geq h_{0}, &\\ \overline{u}(0, x)\geq u_{0}(x), \ \overline{v}(0, x)\geq v_{0}(x), \ \overline{w}(0, x)\geq w_{0}(x), &-h_{0} < x < h_{0}. \end{cases} \end{equation*}

    then the unique solution (u, v, w, g, h) of (3) satisfies

    \begin{align*} &u(t, x)\leq\overline{u}(t, x), \ v(t, x)\leq\overline{v}(t, x), \ w(t, x)\leq\overline{w}(t, x), \\ &g(t)\geq\overline{g}(t), \ h(t)\leq\overline{h}(t) \ \mathit{\text{for}}\ \ 0 < t\leq T, \ g(t)\leq x\leq h(t). \end{align*}

    The proof of Lemma 3.4. Since 2h_{0}<l_\ast , we have \lambda_{p}(\mathcal{L}_{(-h_{0}, h_{0})}^{d_i}+a_i)<0\ (i = 1, 2, 3) . There exists some small \varepsilon>0 such that h^{\ast}: = h_{0}\left(1+\varepsilon\right) satisfies

    \lambda_{p}^i: = \lambda_{p}(\mathcal{L}_{(-h^{\ast}, h^{\ast})}^{d_i}+a_i) < 0.

    Let \phi_i\ (i = 1, 2, 3) be the positive normalized eigenfunction corresponding to \lambda_p^i , namely, \|\phi_i\|_{\infty} = 1 and

    \begin{equation} d_i \int_{-h^{\ast}}^{h^{\ast}}J_i(x-y)\phi_i(y)dy -d_i\phi_i(x)+a_i\phi_i = \lambda_p^i\phi_i, \ x\in[-h^{\ast}, h^{\ast}]. \end{equation} (14)

    Choose positive constants K_i\ (i = 1, 2, 3) large enough such that

    K_1\phi_1(x)\geq u_{0}(x), \ K_2\phi_2(x)\geq v_{0}(x) \ \text{and}\ K_3\phi_3(x)\geq w_{0}(x) \ \text{for}\ x\in[-h_0, h_0].

    Define

    \begin{align*} &\overline{h}(t) = h_{0} \left[1+\varepsilon\left(1-e^{-\delta t}\right)\right], \ \overline{g}(t) = -\overline{h}(t), \ t\geq0, \\ &z_i(t, x) = K_ie^{-\delta t}\phi_i(x), \ t\geq0, \ x\in[\overline{g}(t), \overline{h}(t)], \ i = 1, 2, 3, \end{align*}

    where \delta>0 will be determined later. Clearly h_{0}\leq\overline{h}(t)\leq h^{\ast} .

    For t>0 and x\in(\overline{g}(t), \overline{h}(t)) ,

    \begin{align*} &{z_i}_t-d_i \int_{\overline{g}(t)}^{\overline{h}(t)} J_i(x-y)z_i(t, y)dy +d_iz_i-z_i(a_i-b_iz_i) \end{align*}
    \begin{align*} \geq\ &K_ie^{-\delta t}\Big(-\delta\phi_i(x) -d_i \int_{\overline{g}(t)}^{\overline{h}(t)} J_i(x-y)\phi_i(y)dy+d_i\phi_i-a_i\phi_i\Big)\\ \geq\ &K_ie^{-\delta t}\left(-\delta -\lambda_p^i\right)\phi_i(x)\geq0, \ i = 1, 2, 3, \end{align*}

    if we can choose \delta small enough such that

    \delta\leq\min\{-\lambda_p^1, -\lambda_p^2, -\lambda_p^3\}.

    Moreover, \overline{h}'(t) = h_{0}\varepsilon\delta e^{-\delta t} and

    \begin{align*} &\mu \int_{\overline{g}(t)}^{\overline{h}(t)} \int_{\overline{h}(t)}^{+\infty} [\rho_1J_1(x-y)z_1(t, x)+\rho_2J_2(x-y)z_2(t, x)+ J_3(x-y)z_3(t, x)]dydx\\ \leq\ &2\mu(\rho_1K_1+\rho_2K_2+K_3)e^{-\delta t}h^\ast. \end{align*}

    If

    \mu\leq\frac{h_{0}\varepsilon\delta} {2(\rho_1K_1+\rho_2K_2+K_3)h^{\ast}}: = \mu_{0},

    then we have

    \overline{h}'(t)\geq\mu \int_{\overline{g}(t)}^{\overline{h}(t)} \int_{\overline{h}(t)}^{+\infty} [\rho_1J_1(x-y)z_1(t, x)+\rho_2J_2(x-y)z_2(t, x)+ J_3(x-y)z_3(t, x)]dydx.

    Similarly, we can derive

    \overline{g}'(t)\leq\!-\mu \int_{\overline{g}(t)}^{\overline{h}(t)}\! \int_{-\infty}^{\overline{g}(t)} [\rho_1J_1(x-y)z_1(t, x)+\rho_2J_2(x-y)z_2(t, x)+ J_3(x-y)z_3(t, x)]dydx.

    We may now apply Lemma 3.5 to obtain

    \begin{align*} &u(t, x)\leq z_1(t, x), \ v(t, x)\leq z_2(t, x), \ w(t, x)\leq z_3(t, x), \\ &g(t)\geq\overline{g}(t), \ h(t)\leq\overline{h}(t) \ \text{for}\ t > 0 \ \text{and}\ x\in[g(t), h(t)]. \end{align*}

    It follows that \lim\limits_{t\rightarrow\infty}(h(t)-g(t))\leq \lim\limits_{t\rightarrow\infty}(\overline{h}(t)-\overline{g}(t))\leq 2h^{\ast}<\infty .

    Lemma 3.6. Under the assumptions of Theorem 1.3, if h_0<l_*/2 , then there exists a positive number \mu^0 such that h_\infty-g_\infty = \infty for any \mu>\mu^0 .

    Proof. Consider the following problem

    \begin{equation*} \begin{cases} \underline{w}_{t} = d_3 \int_{\underline{g}(t)}^{\underline{h}(t)} J_3(x-y)\underline{w}(t, y)dy-d_3\underline{w} +\underline{w}(a_3-b_3\underline{w}), &t > 0, \ \underline{g}(t) < x < \underline{h}(t), \\ \underline{w}(t, \underline{g}(t)) = \underline{w}(t, \underline{h}(t)) = 0, &t\geq 0, \\ \underline{g}'(t) = -\mu \int_{\underline{g}(t)}^{\underline{h}(t)} \int_{-\infty}^{\underline{g}(t)} J_3(x-y)\underline{w}(t, x)dydx, &t > 0, \\ \underline{h}'(t) = \mu \int_{\underline{g}(t)}^{\underline{h}(t)} \int_{\underline{h}(t)}^{+\infty} J_3(x-y)\underline{w}(t, x)dydx, &t > 0, \\ -\underline{g}(0) = \underline{h}(0) = h_{0}, &\\ \underline{w}(0, x) = w_{0}(x), &-h_{0} < x < h_{0}. \end{cases} \end{equation*}

    By [3,Theorem 3.1], we have

    w(t, x)\geq\underline{w}(t, x), \ g(t)\leq\underline{g}(t), \ h(t)\geq\underline{h}(t), \ \text{for}\ t > 0, \ x\in(\underline{g}(t), \underline{h}(t)).

    It follows from [3,Theorem 3.13] that there exists some \mu^0 such that \lim\limits_{t\rightarrow\infty} [\underline{h}(t)-\underline{g}(t)] = \infty for any \mu>\mu^0 , and so h_\infty-g_\infty = \infty .

    Then Theorem 1.3 (c) can follow from Lemmas 3.4 and 3.6 by argument in [23,Theorem 5.2]. Next we give the details below for completeness.

    The proof of Theorem 1.3 (c). Define \Sigma^{\ast}: = \{\mu>0:h_{\infty}-g_\infty\leq l_{\ast}\} . By Lemma 3.4, we have (0, \mu_{0}]\subset\Sigma^{\ast} . It follows from Lemma 3.6 that \Sigma^{\ast}\cap[\mu^{0}, \infty) = \emptyset . Therefore, \mu^{\ast}: = \sup\Sigma^{\ast}\in[\mu_{0}, \mu^{0}] . By this definition and Theorem 1.3 (a), we find that h_{\infty}-g_\infty = \infty when \mu>\mu^{\ast} .

    We claim that \mu^{\ast}\in\Sigma^{\ast} . Otherwise h_{\infty}-g_\infty = \infty for \mu = \mu^{\ast} . Hence, we can find T>0 such that h(T)-g(T)>l_{\ast} . To stress the dependence of the solution (u, v, w, g, h) of (3) on \mu , we write (u_{\mu}, v_{\mu}, w_\mu, g_\mu, h_{\mu}) instead of (u, v, w, g, h) . So we have h_{\mu^{\ast}}(T)-g_{\mu^{\ast}}(T)>l_{\ast} . By the continuous dependence of (u_{\mu}, v_{\mu}, w_\mu, g_\mu, h_{\mu}) on \mu , we can find \varepsilon>0 small so that h_{\mu}(T)-g_\mu(T)>l_\ast for \mu\in[\mu^{\ast}-\varepsilon, \mu^{\ast}+\varepsilon] . It follows that for all such \mu ,

    \lim\limits_{t\rightarrow\infty} [h_{\mu}(t)-g_{\mu}(t)] > [h_{\mu}(T)-g_{\mu}(T)] > l_{\ast}.

    This implies that [\mu^{\ast}-\varepsilon, \mu^{\ast}+\varepsilon]\cap\Sigma^{\ast} = \emptyset , and \sup\Sigma^{\ast}\leq\mu^{\ast}-\varepsilon , contradicting to the definition of \mu^{\ast} . This proves our claim.

    Define \Sigma_{\ast}: = \{\nu>0:\nu\geq\mu_{0}\ \text{such that}\ h_{\infty}-g_\infty\leq l_{\ast}\ \text{for all}\ 0<\mu<\nu\} , then \mu_{\ast}: = \sup\Sigma_{\ast}\leq\mu^{\ast} and (0, \mu_{\ast})\subset\Sigma_{\ast} . Similarly to the above, we can prove that \mu_{\ast}\in\Sigma_{\ast} . The proof is completed.

    Finally, we will examine the long-time behaviour of the solution to (3) when h_\infty-g_\infty = \infty . Before proving Theorem 1.4, we first give the following lemma:

    Lemma 3.7. h_\infty = +\infty if and only if g_\infty = -\infty .

    Proof. This follows the idea in the proof of [3,Lemma 3.8]. For example, if g_\infty = -\infty but h_\infty<+\infty , then we may argue as in the proof of [3,Theorem 3.7] to obtain h'(t)\geq \xi_0>0 for all large t , which yields a contradiction.

    The proof of Theorem 1.4. By [3,Theorem 3.9], we have

    \begin{equation} \lim\limits_{t\rightarrow\infty}w(t, x) = \frac{a_3}{b_3} = :C \ \text{locally uniformly for}\ x\in \mathbb R. \end{equation} (15)

    (ⅰ) We will prove it by the following steps.

    Step 1. Let q(t) be the solution of

    \begin{equation*} \begin{cases} q'(t) = q(a_1-b_1q), &t > 0, \\ q(0) = \sup\limits_{x\in \mathbb R}u_0(x).& \end{cases} \end{equation*}

    Then \lim\limits_{t\to\infty}q(t) = a_1/b_1 . By the comparison principle ([3,Lemma 2.2]), we have u(t, x)\leq q(t) for t>0 and x\in[g(t), h(t)] . In view of u(t, x) = 0 for t>0 and x\in \mathbb R\backslash(g(t), h(t)) , we have u(t, x)\leq q(t) for t>0 and x\in \mathbb R . Hence,

    \begin{equation} \limsup\limits_{t\to\infty}u(t, x)\leq\frac{a_1}{b_1} = :\bar A_1 \ \ \text{locally uniformly in}\ \mathbb R. \end{equation} (16)

    Step 2. By (15) and (16), we have

    \limsup\limits_{t\to\infty}[c_2u(t, x)+r_2w(t, x)]\leq c_2\bar A_1+r_2C \ \ \text{locally uniformly in}\ \mathbb R.

    By the condition a_2>c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3} , we have

    a_2-c_2\bar A_1-r_2C = a_2-c_2\frac{a_1}{b_1}-r_2\frac{a_3}{b_3} > 0.

    It follows from above two facts, and [6,Lemma 3.14] that

    \begin{equation} \liminf\limits_{t\to\infty}v(t, x)\geq \frac{a_2-c_2\bar A_1-r_2C}{b_2} = :\underline B_1 \ \text{locally uniformly in}\ \mathbb R. \end{equation} (17)

    Step 3. By (15) and (17), we have

    \limsup\limits_{t\to\infty}[c_1v(t, x)+r_1w(t, x)]\geq c_1\underline{B}_1+r_1C \ \ \text{locally uniformly in}\ \mathbb R.

    The condition a_1>c_1\frac{a_2}{b_2}+r_1\frac{a_3}{b_3} implies

    \begin{align*} &a_1-c_1\underline B_1-r_1C = a_1-c_1\frac{a_2-c_2\bar A_1-r_2C}{b_2}-r_1C \geq a_1-c_1\frac{a_2}{b_2}-r_1\frac{a_3}{b_3} > 0. \end{align*}

    These two facts and [6,Lemma 3.14] allow us to derive

    \begin{equation} \limsup\limits_{t\to\infty}u(t, x)\leq \frac{a_1-c_1\underline B_1-r_1C}{b_1} = :\bar A_2 \ \ \text{locally uniformly in}\ \mathbb R. \end{equation} (18)

    Step 4. By (15) and (18), we have

    \limsup\limits_{t\to\infty}[c_2u(t, x)+r_2w(t, x)]\leq c_2\bar A_2+r_2C \ \ \text{locally uniformly in}\ \mathbb R.

    Furthermore, the condition a_2>c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3} implies

    a_2-c_2\bar A_2-r_2C = a_2-c_2\frac{a_1-c_1\underline B_1-r_1C}{b_1}-r_2C \geq a_2-c_2\frac{a_1}{b_1}-r_2\frac{a_3}{b_3} > 0.

    Similar to the above,

    \liminf\limits_{t\to\infty}v(t, x)\geq (a_2-c_2\bar A_2-r_2C)/{b_2} = :\underline B_2 \ \ \text{locally uniformly in}\ \mathbb R.

    Step 5. Repeating the above procedure, we can find two sequences \bar A_i and \underline B_i such that

    \limsup\limits_{t\to\infty}u(t, x)\leq\bar A_i, \ \ \ \liminf\limits_{t\to\infty}v(t, x)\geq\underline B_i \ \ \text{locally uniformly in}\ \mathbb R,

    and

    \bar A_{i+1} = (a_1-c_1\underline B_i-r_1C)/{b_1}, \ \ \ \underline B_i = (a_2-c_2\bar A_i-r_2C)/{b_2}, \ \ \ i = 1, 2, \cdots.

    Let

    p: = \frac{a_1-r_1C}{b_1}-\frac{c_1(a_2-r_2C)}{b_1b_2}, \; \ \ \ q: = \frac{c_1c_2}{b_1b_2}.

    Then p>0 by a_1-c_1\frac{a_2}{b_2}-r_1\frac{a_3}{b_3}>0 , 0<q<1 by a_1>c_1\frac{a_2}{b_2} and a_2>c_2\frac{a_1}{b_1} . By direct calculation,

    \bar A_{i+1} = p+q\bar A_i, \ \ \ i = 1, 2, \cdots.

    From \bar A_2<\bar A_1 and the above iteration formula, we immediately obtain

    0 < \bar A_{i+1} < \bar A_i, \;\; \ i = 1, 2, \cdots,

    from which it easily follows that

    \lim\limits_{i\to\infty}\bar A_i = \frac{b_2(a_1b_3-r_1a_3)-c_1(a_2b_3-r_2a_3)}{b_3(b_1b_2-c_1c_2)},
    \lim\limits_{i\to\infty}\underline B_i = \frac{-c_2(a_1b_3-r_1a_3)+b_1(a_2b_3-r_2a_3)}{b_3(b_1b_2-c_1c_2)}.

    Thus we have

    \limsup\limits_{t\to\infty}u(t, x)\leq \frac{b_2(a_1b_3-r_1a_3)-c_1(a_2b_3-r_2a_3)}{b_3(b_1b_2-c_1c_2)} \ \text{locally uniformly in}\ \mathbb R.
    \liminf\limits_{t\to\infty}v(t, x)\geq \frac{-c_2(a_1b_3-r_1a_3)+b_1(a_2b_3-r_2a_3)}{b_3(b_1b_2-c_1c_2)} \ \text{locally uniformly in}\ \mathbb R.

    Similarly, we can show

    \liminf\limits_{t\to\infty}u(t, x)\geq \frac{b_2(a_1b_3-r_1a_3)-c_1(a_2b_3-r_2a_3)}{b_3(b_1b_2-c_1c_2)} \ \text{locally uniformly in}\ \mathbb R.
    \limsup\limits_{t\to\infty}v(t, x)\leq \frac{-c_2(a_1b_3-r_1a_3)+b_1(a_2b_3-r_2a_3)}{b_3(b_1b_2-c_1c_2)} \ \text{locally uniformly in}\ \mathbb R.

    Thus, (ⅰ) is proved.

    (ⅱ) By Steps 1 and 2 in (ⅰ), we also have

    \begin{equation*} \liminf\limits_{t\to\infty}v(t, x)\geq \frac{a_2-c_2\bar A_1-r_2C}{b_2} = :\underline B_1 \ \text{locally uniformly in}\ \mathbb R. \end{equation*}

    It follows from this fact and (15) that

    \limsup\limits_{t\to\infty}[c_1v(t, x)+r_1w(t, x)]\geq c_1\underline{B}_1+r_1C \ \ \text{locally uniformly in}\ \mathbb R.

    The condition a_1+\frac{c_1}{b_2} \left(c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3}\right)\leq c_1\frac{a_2}{b_2}+r_1\frac{a_3}{b_3} implies

    \begin{align*} &a_1-c_1\underline B_1-r_1C = a_1-c_1\frac{a_2-c_2\bar A_1-r_2C}{b_2}-r_1C\\ = \ &a_1+\frac{c_1}{b_2} \Big(c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3}\Big)- c_1\frac{a_2}{b_2}-r_1\frac{a_3}{b_3}\leq0. \end{align*}

    These two facts and [6,Lemma 3.14] allow us to derive

    \begin{equation*} \limsup\limits_{t\to\infty}u(t, x)\leq0 \ \text{locally uniformly in}\ \mathbb R. \end{equation*}

    Since

    \liminf\limits_{t\to\infty}u(t, x)\geq0 \ \text{locally uniformly in}\ \mathbb R,

    we have

    \lim\limits_{t\to\infty}u(t, x) = 0 \ \text{locally uniformly in}\ \mathbb R.

    It follows from this, (15) and [6,Lemma 3.14] that

    \frac{a_2b_3-r_2a_3}{b_2b_3}\leq\liminf\limits_{t\to\infty}v(t, x)\leq\limsup\limits_{t\to\infty}v(t, x) \leq\frac{a_2b_3-r_2a_3}{b_2b_3} \ \text{locally uniformly in}\ \mathbb R,

    and so

    \lim\limits_{t\to\infty}v(t, x) = \frac{a_2b_3-r_2a_3}{b_2b_3} \ \text{locally uniformly in}\ \mathbb R.

    We have proved (ⅱ).

    (ⅲ) This conclusion can be proved by the same arguments in (ⅱ).

    (ⅳ) By (15), we have

    \limsup\limits_{t\to\infty}[c_1v(t, x)+r_1w(t, x)]\geq r_1C \ \text{locally uniformly in}\ \mathbb R.

    The condition a_1\leq r_1\frac{a_3}{b_3} implies

    \begin{align*} &a_1-r_1C = a_1-r_1\frac{a_3}{b_3}\leq0. \end{align*}

    These two facts and [6,Lemma 3.14] allow us to derive

    \begin{equation*} \limsup\limits_{t\to\infty}u(t, x)\leq0 \ \text{locally uniformly in}\ \mathbb R, \end{equation*}

    and so

    \lim\limits_{t\to\infty}u(t, x) = 0 \ \text{locally uniformly in}\ \mathbb R.

    Similarly, we have

    \lim\limits_{t\to\infty}v(t, x) = 0 \ \text{locally uniformly in}\ \mathbb R.

    Then (ⅳ) has been proved.

    In this paper, we study a free boundary problem with nonlocal diffusion describing information diffusion in online social networks. This system consists of three equations representing three pieces of information propagating via the internet and competing for influence among users. We obtain the criteria for information spreading and vanishing. If the diffusion rate of any piece of information is small, i.e., d_1\leq a_1 or d_2\leq a_2 or d_3\leq a_3 , information will always spread. But when the diffusion rates of three pieces of information are all large, i.e., d_i>a_i\ (i = 1, 2, 3) , whether information spread or vanish depends on the initial data. If the initial spreading area [-h_0, h_0] is within the critical size, i.e., h_0<l^*/2 , information spread or vanish depending on the size of the expanding capacity \mu , namely, vanishing happens with small expanding capability and spreading happens with large expanding capability. Regardless of the expanding capability, spreading always occurs if the initial spreading area is beyond the critical size. We find that the result of the nonlocal diffusion model (3) is different from the local diffusion model in [15].

    When spreading happens, the longtime behavior of the solution is obtained in Theorem 1.4, which is similar to the result of local diffusion model studied in [15]. According to Theorem 1.4, we can choose suitable official information to control rumor propagation in social networks, namely, we can change the value of a_3 and b_3 by choosing suitable official information.

    For local diffusion model (2), the result in [15] showed the spreading has a finite speed when spreading happens. However, what will happen for the nonlocal diffusion model (3)? Very recently, Du, Li and Zhou [4] investigated the spreading speed of the nonlocal model in [3] and proved that the spreading may or may not have a finite speed, depending on whether a certain condition is satisfied by the kernel function J in the nonlocal diffusion term. This contrasts sharply to the local model of [5], where the spreading has finite speed whenever spreading happens. Since (3) consists of three equations, we expect a more complex result for (3), which will be considered in a future work.



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  • This article has been cited by:

    1. Phuong Le, Longtime behavior of a free boundary model with nonlocal diffusion in online social networks, 2025, 6, 2662-2963, 10.1007/s42985-025-00324-3
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