The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks

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, $\mu$ 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].

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

where $d_i\ (i = 1, 2, 3)$, $a_i\ (i = 1, 2, 3)$, $b_i\ (i = 1, 2, 3)$, $c_i\ (i = 1, 2)$, $r_i\ (i = 1, 2)$, $\rho_i\ (i = 1, 2)$, $\mu$ and $h_0$ are given positive constants. The initial functions $u_0(x), \ v_0(x)$ and $w_0(x)$ belong to

where $[-h_0, h_0]$ represents the initial range of the information. Assumed that $u, v, w$ are identically $0$ for $x\in \mathbb R\backslash[g(t), h(t)]$, and the kernel function $J_i: \mathbb R\rightarrow \mathbb R\ (i = 1, 2, 3)$ is continuous and nonnegative, and have the properties

$\textbf{(J)}$ $J(0)>0, \ \int_{ \mathbb R}J(x)dx = 1$, $J$ is symmetric, $\sup\limits_{ \mathbb R}J<\infty$.

The main results of this paper are the following theorems:

Theorem 1.1 (Global existence and uniqueness). Suppose that $J_i (i = 1, 2, 3)$ satisfies (J). Then for any given $h_0>0$ and $u_0(x), v_0(x), w_0(x)$ belonging to $\mathscr{X}(h_0)$, 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 $J_1(x)>0$, $J_2(x)>0$ in $\mathbb{R}$, then one of the following alternatives must happen:

${\rm(i)}$ $\rm Spreading:$ $\lim\limits_{t\rightarrow\infty} [h(t)-g(t)] = \infty$.

${\rm(ii)}$ $\rm Vanishing:$ $\lim\limits_{t\rightarrow\infty} (g(t), h(t)) = (g_\infty, h_\infty)$ is a finite interval,

and

Theorem 1.3 (Spreading-vanishing criteria). Assume that $J_i\ (i = 1, 2, 3)$ satisfies (J), and $J_1(x)>0$, $J_2(x)>0$ in $\mathbb{R}$. Then the dichotomy in Theorem 1.2 can be determined as follows:

${\rm(i)}$ If $a_1\geq d_1$ or $a_2\geq d_2$ or $a_3\geq d_3$, then necessarily $h_\infty-g_\infty = \infty$.

${\rm(ii)}$ If $a_i<d_i \mathit{\text{for}} \ \ i = 1, 2, 3$, then

${\rm (a)}$ If $h_\infty-g_\infty<\infty$, then $h_\infty-g_\infty\leq l_*$.

${\rm (b)}$ If $h_0\geq l_*/2$, then $h_\infty-g_\infty = \infty$.

${\rm (c)}$ If $h_0<l_*/2$, then there exist two positive numbers $\mu^*\geq\mu_*>0$ such that $h_\infty-g_\infty<\infty$ when $0<\mu\leq\mu_*$ and $\mu = \mu^*$, and $h_\infty-g_\infty = \infty$ when $\mu>\mu^*$,

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 $\lim\limits_{t\to\infty}[h(t)-g(t)] = \infty$. The following conclusions hold:

$\rm (i)$ If $a_1>c_1\frac{a_2}{b_2}+r_1\frac{a_3}{b_3}$ and $a_2>c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3}$, then

locally uniformly for $x\in \mathbb R$.

$\rm (ii)$ If $a_1+\frac{c_1}{b_2} \left(c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3}\right)\leq c_1\frac{a_2}{b_2}+r_1\frac{a_3}{b_3}$ and $a_2>c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3}$, then

$\rm (iii)$ If $a_1>c_1\frac{a_2}{b_2}+r_1\frac{a_3}{b_3}$ and $a_2+\frac{c_2}{b_1}\left(c_1\frac{a_2}{b_2}+r_1\frac{a_3}{b_3}\right)\leq c_2\frac{a_1}{b_1}+r_2\frac{a_3}{b_3}$, then

$\rm (iv)$ If $a_1\leq r_1\frac{a_3}{b_3}$ and $a_2\leq r_2\frac{a_3}{b_3}$, 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 $\mu$ are both sufficiently small. However, for (3), Theorem 1.3 indicates that when $d_1\leq a_1$ or $d_2\leq a_2$ or $d_3\leq a_3$, spreading always happens no mater how small $h_0$ and $\mu$ 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 $J_3$ satisfies (J), $f_3(w): = w(a_3-b_3w)$ satisfies (f1) and (f2) in [3], and $w_0(x)$ belongs to $\mathscr{X}(h_0)$ for any $h_0>0$. For any given $T>0$ and $(g, h)\in G_T\times H_T$, it follows from [3,Lemma 2.3] that the following problem

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

For such $w(t, x)$, it is easy to check that $f_1(t, x, u, v): = u(a_1-b_1u-c_1v+r_1w(t, x))$ and $f_2(t, x, u, v): = v(a_2-b_2v-c_2u+r_2w(t, x))$ satisfy (f), (f1) and (f2) in [6]. For $(g, h)$ given above, it follows from [6,Lemma 2.3] that the following problem

has a unique solution $(u, v)$ and

For $(u, v, w, g, h)$ above, we define $(\widetilde{g}, \widetilde{h})$ for $t\in[0, T]$ by

Since $J_i\ (i = 1, 2, 3)$ satisfies (J), there exist constants $\epsilon_0\in(0, h_0/4)$ and $\delta_0$ such that

Let

then

Using this we can follow the corresponding arguments of [6] to show that, for some sufficiently small $T_0 = T_0(\mu, A_1, A_2, A_3, h_0, \epsilon_0, \rho_1, \rho_2, J)>0$ and any $T\in (0, T_0]$,

where

Let

and define the mapping

Then the above analysis indicates that

Next, we will show that $\mathcal{F}$ is a contraction mapping on $\Sigma_T$ for sufficiently small $T\in(0, T_0]$. For any given $(g_i, h_i)\in\Sigma_T\ (i = 1, 2)$, denote

Let

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 $T_1\in(0, T_0)$ such that, for $T\leq T_1$,

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 $T_2\in(0, T_1)$ such that, for $T\leq T_2$,

Then we have, for $T\leq T_2$,

where

This shows that if we choose $\widetilde{T}$ such that

then, for $T\in(0, \widetilde{T}]$,

and so $\mathcal{F}$ is a contraction mapping on $\Sigma_T$. Hence $\mathcal{F}$ has a unique fixed point $(g, h)$ in $\Sigma_T$, which gives a nonnegative solution $(u, v, w, g, h)$ of (3) for $t\in(0, T]$. 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 $\beta_1, \beta_2, \beta_3>0$ be constants. Suppose that $g, \, h\in C^1([0, \infty))$, $g(0)<h(0)$, $g'(t)\leq0$, $h'(t)\geq0$ and $w_i, \, w_{it}\in C(D_\infty)\cap L^\infty(D_\infty)$ for $i = 1, 2, 3$, where $D_\infty = \{t>0, \ g(t)<x<h(t)\}$. If $(w_1, w_2, w_3, g, h)$ satisfies

and

then

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 $\mathbb{R}$. Suppose that $g, h\in C^1([0, \infty))$, $g(0)<h(0)$, $g'(t)\leq0$, $h'(t)\geq0$, and (5) holds. If $(w, g, h)$ satisfies, for some positive constants $\beta$ and $M$,

and $\lim\limits_{t\to\infty}h'(t) = 0$, then

We define the operator $\mathcal{L}_{\Omega}^{d_i}+\beta : C(\overline{\Omega})\rightarrow C(\overline{\Omega})$ by

where $\Omega$ is an open bounded interval in $\mathbb R$, and $\beta\in C(\overline{\Omega})$. The generalized principal eigenvalue of $\mathcal{L}_{\Omega}^{d_i}+\beta$ is given by

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

Lemma 3.3. Assume that $J_i\ (i = 1, 2, 3)$ satisfies (J), $J_i(x)>0\ (i = 1, 2)$ in $\mathbb{R}$. Let $(u, v, w, g, h)$ be the unique solution of (3). If $h_\infty-g_\infty<\infty$, then

moreover,

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

In the following, we only prove

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

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

Define

and

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

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

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

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

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

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

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

We now make use of the identity

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

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

Therefore we can find $T>0$ large so that

It then follows from the equation satisfied by $u$ that

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

Moreover, for such $\epsilon$, it follows from

that there exists $T_\epsilon$ such that

and

Then

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

Then, for any $\delta>0$,

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

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],

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.

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

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

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

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

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

Define

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

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

if we can choose $\delta$ small enough such that

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

If

then we have

Similarly, we can derive

We may now apply Lemma 3.5 to obtain

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

By [3,Theorem 3.1], we have

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 $\mu$,

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.

3.2.   Long-time behaviour in the case of spreading

Finally, we will examine the long-time behaviour of the solution to (3) when $h_\infty-g_\infty = \infty$. 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

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

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

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,

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

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

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

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

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

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

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

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

Similar to the above,

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

and

Let

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,

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

from which it easily follows that

Thus we have

Similarly, we can show

Thus, (ⅰ) is proved.

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

It follows from this fact and (15) that

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

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

Since

we have

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

and so

We have proved (ⅱ).

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

(ⅳ) By (15), we have

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

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

and so

Similarly, we have

Then (ⅳ) has been proved.

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., $d_1\leq a_1$ 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.