THE LONGTIME BEHAVIOR OF THE MODEL WITH NONLOCAL DIFFUSION AND FREE BOUNDARIES IN ONLINE SOCIAL NETWORKS

In this paper we consider a free boundary problem with nonlocal diffusion describing information diffusion in online social networks. This model can be viewed as a nonlocal version of the free boundary problem studied by Ren et al. (Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019) 1843–1865). We first show that this problem has a unique solution for all t > 0, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy. We also obtain sharp criteria for spreading and vanishing, and show that the spreading always happen if the diffusion rate of any one of the information is small, which is very different from the local diffusion model.


1.
Introduction. 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: 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, 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 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, d i (i = 1, 2, 3) is the diffusion rates, a i (i = 1, 2, 3) is the intrinsic growth rates, 1/b i (i = 1, 2, 3) is the carrying capacities, c i (i = 1, 2) and r i (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 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].
(iv) If a 1 ≤ r 1 a3 b3 and a 2 ≤ r 2 a3 b3 , then Remark 1. Note that for the corresponding local diffusion model in [15], no matter how small the diffusion coefficient d i is, vanishing can always happen if h 0 and µ are both sufficiently small. However, for (3), Theorem 1.3 indicates that when d 1 ≤ a 1 or d 2 ≤ a 2 or d 3 ≤ a 3 , spreading always happens no mater how small h 0 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.

2.
Global existence and uniqueness. For convenience, we first introduce some notations. For given T > 0, define 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 admits a unique solution w(t, x), and ) satisfy (f ), (f1) and (f2) in [6]. For (g, h) given above, it follows from [6, Lemma 2.3] that the following problem has a unique solution (u, v) and Since Using this we can follow the corresponding arguments of [6] to show that, for some and define the mapping F(g, h) = ( g, h).
Then the above analysis indicates that Next, we will show that F is a contraction mapping on Σ T for sufficiently small T ∈ (0, T 0 ]. For any given (g i , h i ) ∈ Σ T (i = 1, 2), denote Then we can follow the approach of Step 2 in the proof of [3, Theorem 2.1] to show We can apply the same argument as the step 2 in the proof of [6, Theorem 2.1] to obtain that there exist C 1 and T 1 ∈ (0, T 0 ) such that, for T ≤ T 1 , Meanwhile, by using the same argument as the step 2 in the proof of [3, Theorem 2.1] to obtain that there exist C 2 and T 2 ∈ (0, T 1 ) such that, for T ≤ T 2 , Then we have, for T ≤ T 2 , This shows that if we choose T such that then, for T ∈ (0, T ], and so F is a contraction mapping on Σ T . Hence F has a unique fixed point (g, h) in Σ T , 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.
3. Spreading and vanishing. Since u, v and w are positive in D T , we have h (t) > 0 and g (t) < 0 for t > 0. Thus we can define Clearly we have either 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.
3.1. Criteria for vanishing and spreading. Before analysing the vanishing phenomenon, we first give some lemmas.
Lemma 3.1. Let the condition (J) hold for the kernel functions J i (i = 1, 2, 3), and and lim 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.  (5) holds. If (w, g, h) satisfies, for some positive constants β and M , We define the operator L di Ω + β : C(Ω) → C(Ω) by where Ω is an open bounded interval in R, and β ∈ C(Ω). The generalized principal eigenvalue of L di Ω + β is given by Then we will use the techniques in [6, Theorem 3.3] to give the vanishing result.
The conclusion for v can be obtained similarly, so we omit here. By the same arguments in [6], we can have

u(t, x)
and By the arguments in [6], the following claim holds By the above stated properties of M (t), there exists a sequence t n > 0 increasing to ∞ as n → ∞, and ξ n ∈ {ξ(t n ), ξ(t n )} such that lim n→∞ u(t n , ξ n ) = σ * , lim n→∞ u t (t n , ξ n ) = 0.

By
We now make use of the identity with (t, x) = (t n , ξ n ). Letting n → ∞, we obtain It follows that a 1 > d 1 . We show next that this leads to a contradiction.
Moreover, for such , it follows from that there exists T such that Then Let φ(x) be the corresponding normalized eigenfunction of λ p (L d1 (g∞+ ,h∞− ) + a 1 − 2 (c 1 + r 1 )), namely, φ ∞ = 1 and Then, for any δ > 0, then we can use [3, Lemma 3.3] and a simple comparison argument to obtain This is a contradiction to lim 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 Proof. Arguing indirectly we assume that h ∞ − g ∞ < ∞ and a i ≥ d i for some i ∈ {1, 2, 3}. Thanks to [3,Proposition 3.4], This is a contradiction to Lemma 3.3.
Hence, Theorem 1.3 (i) has been proved. We next consider the case that In this case, it follows from [3, Proposition 3.4] that there exists l i (i = 1, 2, 3) such that Define 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. 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.
For t > 0 and x ∈ (g(t), h(t)), Similarly, we can derive We may now apply Lemma 3.5 to obtain Proof. Consider the following problem

It follows that lim
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.

3.2.
Long-time behaviour in the case of spreading. Finally, we will examine the long-time behaviour of the solution to (3) when h ∞ − g ∞ = ∞. Before proving Theorem 1.4, we first give the following lemma: (i) We will prove it by the following steps.
4. Discussion. 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 ≤ a 1 or d 2 ≤ a 2 or d 3 ≤ 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 µ, 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.