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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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(1−u/K),t>1, l<x<L,u(1,x)=u0(x),l≤x≤L,ux(t,l)=0, ux(t,L)=0,t>1, |
where
In above system,
{ut=duxx+r(t)u(1−u/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),0≤x≤h0. | (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(a1−b1u−c1v−r1w),t>0, 0<x<h(t),vt=d2vxx+v(a2−b2v−c2u−r2w),t>0, 0<x<h(t),wt=d3wxx+w(a3−b3w),t>0, 0<x<h(t),ux(t,0)=vx(t,0)=wx(t,0)=0,t≥0,u(t,h(t))=v(t,h(t))=w(t,h(t))=0,t≥0,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
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
d∫RJ(x−y)u(t,y)dy−du(t,x), |
which can capture short-range as well as long-range factors in the dispersal by choosing the kernel function
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=d1∫h(t)g(t)J1(x−y)u(t,y)dy−d1u+u(a1−b1u−c1v−r1w),t>0, g(t)<x<h(t),vt=d2∫h(t)g(t)J2(x−y)v(t,y)dy−d2v+v(a2−b2v−c2u−r2w),t>0, g(t)<x<h(t),wt=d3∫h(t)g(t)J3(x−y)w(t,y)dy−d3w+w(a3−b3w),t>0, g(t)<x<h(t),u(t,x)=v(t,x)=w(t,x)=0,t≥0, x=g(t) or h(t),g′(t)=−μ∫h(t)g(t)∫g(t)−∞[ρ1J1(x−y)u(t,x)+ρ2J2(x−y)v(t,x)+J3(x−y)w(t,x)]dydx,t>0,h′(t)=μ∫h(t)g(t)∫+∞h(t)[ρ1J1(x−y)u(t,x)+ρ2J2(x−y)v(t,x)+J3(x−y)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
X(h0):={u0∈C([−h0,h0]) : u0(±h0)=0, u0>0 in (−h0,h0)}, |
where
The main results of this paper are the following theorems:
Theorem 1.1 (Global existence and uniqueness). Suppose that
Theorem 1.2 (Spreading-vanishing dichotomy). Let the conditions of Theorem 1.1 hold and
limt→∞maxg(t)≤x≤h(t)u(t,x)=0, limt→∞maxg(t)≤x≤h(t)v(t,x)=0 |
and
limt→∞maxg(t)≤x≤h(t)w(t,x)=0. |
Theorem 1.3 (Spreading-vanishing criteria). Assume that
where
Theorem 1.4 (Asymptotic limit). Let
limt→∞(u(t,x),v(t,x),w(t,x))= (b2(a1b3−r1a3)−c1(a2b3−r2a3)b3(b1b2−c1c2),−c2(a1b3−r1a3)+b1(a2b3−r2a3)b3(b1b2−c1c2),a3b3) |
locally uniformly for
limt→∞(u(t,x),v(t,x),w(t,x))=(0,a2b3−r2a3b2b3,a3b3) locally uniformly for x∈R. |
limt→∞(u(t,x),v(t,x),w(t,x))=(a1b3−r1a3b1b3,0,a3b3) locally uniformly for x∈R. |
limt→∞(u(t,x),v(t,x),w(t,x))=(0,0,a3b3) locally uniformly for x∈R. |
Remark 1. Note that for the corresponding local diffusion model in [15], no matter how small the diffusion coefficient
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
For convenience, we first introduce some notations. For given
HT:={h∈C1([0,T]):h(0)=h0,h(t) is strictly increasing},GT:={g∈C1([0,T]):−g∈HT},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
{wt=d3∫h(t)g(t)J3(x−y)w(t,y)dy−d3w+w(a3−b3w),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),−h0≤x≤h0 |
admits a unique solution
0<w(t,x)≤max{‖w0‖∞,a3/b3}=:A3 in DT. |
For such
{ut=d1∫h(t)g(t)J1(x−y)u(t,y)dy−d1u+u(a1−b1u−c1v−r1w(t,x)),0<t<T, g(t)<x<h(t),vt=d2∫h(t)g(t)J2(x−y)v(t,y)dy−d2v+v(a2−b2v−c2u−r2w(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),−h0≤x≤h0 |
has a unique solution
0<u≤max{‖u0‖∞,a1/b1}=:A1 in DT, |
0<v≤max{‖v0‖∞,a2/b2}=:A2 in DT. |
For
{˜g(t):=−h0−μ∫t0∫h(τ)g(τ)∫g(τ)−∞[ρ1J1(x−y)u(τ,x)+ρ2J2(x−y)v(τ,x)+J3(x−y)w(τ,x)]dydxdτ,˜h(t):=h0+μ∫t0∫h(τ)g(τ)∫+∞h(τ)[ρ1J1(x−y)u(τ,x)+ρ2J2(x−y)v(τ,x)+J3(x−y)w(τ,x)]dydxdτ. |
Since
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(a1−b1u−c1v−r1w)≥−(b1A1+c1A2+r1A3)u≥−Lu,f2(u,v,w)=v(a2−b2v−c2u−r2w)≥−(b2A2+c2A1+r2A3)v≥−Lv,f3(w)=w(a3−b3w)≥−b3A3w≥−Lw. |
Using this we can follow the corresponding arguments of [6] to show that, for some sufficiently small
sup0≤t1<t2≤T˜g(t2)−˜g(t1)t2−t1≤−˜σ0, inf0≤t1<t2≤T˜h(t2)−˜h(t1)t2−t1≥σ0, |
˜h(t)−˜g(t)≤2h0+ϵ04 for t∈[0,T], |
where
˜σ0=14ϵ0δ0μe−(d1+d2+d3+L)T0∫−h0+ϵ04−h0 [ρ1u0(x)+ρ2v0(x)+w0(x)]dx, |
σ0=14ϵ0δ0μe−(d1+d2+d3+L)T0∫h0h0−ϵ04[ρ1u0(x)+ρ2v0(x)+w0(x)]dx. |
Let
ΣT:={(g,h)∈GT×HT : sup0≤t1<t2≤Tg(t2)−g(t1)t2−t1≤−˜σ0,inf0≤t1<t2≤Th(t2)−h(t1)t2−t1≥σ0, 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
(˜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)|≤ 6h0μρ1T‖U‖C([0,T]×R)+6h0μρ2T‖V‖C([0,T]×R)+6h0μT‖W‖C([0,T]×R)+3Tμ(ρ1A1+ρ2A2+A3)[‖g1−g2‖C([0,T])+‖h1−h2‖C([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
max{‖U‖C([0,T]×R), ‖V‖C([0,T]×R)}≤C1[‖g1−g2‖C([0,T])+‖h1−h2‖C([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
‖W‖C([0,T]×R)≤C2[‖g1−g2‖C([0,T])+‖h1−h2‖C([0,T])]. |
Then we have, for
|˜g1(t)−˜g2(t)|+|˜h1(t)−˜h2(t)|≤ C3T[‖g1−g2‖C([0,T])+‖h1−h2‖C([0,T])], |
where
C3=6h0μρ1C1+6h0μρ2C1+6h0μC2+3μ(ρ1A1+ρ2A2+A3). |
This shows that if we choose
0<˜T≤min{T2,12C3}, |
then, for
|˜g1(t)−˜g2(t)|+|˜h1(t)−˜h2(t)|≤ 12[‖g1−g2‖C([0,T])+‖h1−h2‖C([0,T])] |
and so
Since
limt→∞g(t)=g∞∈[−∞,−h0), limt→∞h(t)=h∞∈(h0,∞]. |
Clearly we have either
(i) h∞−g∞<∞, or (ii) h∞−g∞=∞. |
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
Before analysing the vanishing phenomenon, we first give some lemmas.
Lemma 3.1. Let the condition (J) hold for the kernel functions
{h′(t)=3∑i=1βi∫h(t)g(t)∫∞h(t)Ji(x−y)wi(t,x)dydx,t≥0,g′(t)=−3∑i=1βi∫h(t)g(t)∫g(t)−∞Ji(x−y)wi(t,x)dydx,t≥0, | (4) |
and
limt→∞h(t)−limt→∞g(t)<∞, | (5) |
then
limt→∞g′(t)=limt→∞h′(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
0≤w≤M in D∞, w(t,g(t))=w(t,h(t))=0, ∀ t≥0, |
h′(t)≥β∫h(t)g(t)∫∞h(t)J(x−y)w(t,x)dydx, ∀ t>0, |
and
limt→∞∫h(t)g(t)w(t,x)dx=0, ∫∞0∫h(t)g(t)w(t,x)dxdt<∞. |
We define the operator
(LdiΩ+β)[ϕ](x):=di∫ΩJi(x−y)ϕ(y)dy−diϕ(x)+β(x)ϕ(x), |
where
λp(LdiΩ+β):=inf{λ∈R:(LdiΩ+β)[ϕ]≤λϕ in Ω for some ϕ∈C(¯Ω), ϕ>0}. |
Then we will use the techniques in [6,Theorem 3.3] to give the vanishing result.
Lemma 3.3. Assume that
limt→∞maxg(t)≤x≤h(t)u(t,x)=limt→∞maxg(t)≤x≤h(t)v(t,x)=limt→∞maxg(t)≤x≤h(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
limt→∞maxg(t)≤x≤h(t)w(t,x)=0 and λp(Ld3(g∞,h∞)+a3)≤0. |
In the following, we only prove
limt→∞maxg(t)≤x≤h(t)u(t,x)=0 and λp(Ld1(g∞,h∞)+a1)≤0. | (8) |
The conclusion for
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
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
Now we are ready to show that
\begin{equation} \sigma^\ast: = \limsup\limits_{t\rightarrow\infty}M(t)\in(0, \infty). \end{equation} | (10) |
By the above stated properties of
\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
\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} \int_{g(t)}^{h(t)}u(t, y)dy = 0. |
Since
\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
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
Indeed, by (9), there exists
\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
-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_t(t, x_0)\geq \frac{a_1-d_1}{2}u(t, x_0) \ \mbox{ for } t\geq T, |
which implies
In the following we prove the second conclusion of (8). Suppose on the contrary 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
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
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
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
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
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
Then Theorem 1.2 can be obtained by Lemma 3.3 directly.
Corollary 1. Suppose that
Proof. Arguing indirectly we assume that
\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
\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
Lemma 3.4. Under the assumptions of Theorem 1.3, if
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
\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
\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
\lambda_{p}^i: = \lambda_{p}(\mathcal{L}_{(-h^{\ast}, h^{\ast})}^{d_i}+a_i) < 0. |
Let
\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_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
For
\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\leq\min\{-\lambda_p^1, -\lambda_p^2, -\lambda_p^3\}. |
Moreover,
\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
Lemma 3.6. Under the assumptions of Theorem 1.3, if
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
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
We claim that
\lim\limits_{t\rightarrow\infty} [h_{\mu}(t)-g_{\mu}(t)] > [h_{\mu}(T)-g_{\mu}(T)] > l_{\ast}. |
This implies that
Define
Finally, we will examine the long-time behaviour of the solution to (3) when
Lemma 3.7.
Proof. This follows the idea in the proof of [3,Lemma 3.8]. For example, if
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
\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
\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\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
\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\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
\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
\bar A_{i+1} = p+q\bar A_i, \ \ \ i = 1, 2, \cdots. |
From
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
\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
\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.,
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
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
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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 |