Describe A Class Of Free Boundary Problem Of Information Dissemination

With the developing of technology, people gradually entered a new society-the digital information society. The networks become an indispensable tool of information dissemination in this new society network. With the extensive application of the network, peopele have been interested in how the information spreads through the network. In order to understand how the information transmit through the Internet, many mathematical models have emerged to model the information diffusion in recent decades.Research efforts on understanding information diffusion have a significant impact on real life applications such as product marketing, political online campaign, etc. Extensive investigations have been made to understand network structure, user interactions, and traffic properties, and to study the characteristics of information diffusion. Mathematical modeling has played an increasingly important role in understanding information diffusion in online social networks. Most existing models of information diffusion on online social networks have concentrated on the temporal dimension. Recently a Diffusive Logistic(DL) model was proposed by Wang to study the process of information diffusion in online social networks over time and position. The disentation is dovted to the frontier of the diffusion and discuss the corresponding free boundary problem.In Section1, the background of information diffusion in online social netwoks is first presented, we will show how to establish the Logistic diffusion model and the main results are also given.In Section2, we first show that the solution of (1.3) is global and unique, and the free boundary x=h(t) is increasing. Then we present the comparison principle. Finally, we use the comparison principle to give the upper bound of the solution.In Section3, we show that the information either lasts forever or suspends in finite time, that is, if then h∞=∞and lim(?) u(t,·)=K in any bounded domain, which means the information diffusion lasts forever. On the other hand, if h∞<∞, then and lim(?)=0uniformly. In the other words, the information vanishes in finite time.In Section4, by constructing an upper solution we prove that if λ is sufficiently small, the information vanishing must occur. Then we show that there exists a threshold λ*, which is dependent on φ∈∑(h0), such that when λ>λ*, the information with the initial data u0=λφ travels in the whole distance. Otherwise, the information vanishing happens.In Section5, we demonstrate that if the information spreading happens, the expanding front x=h(t) moves at a constant speed k0, which is determined by an elliptic equation derived from the free boundary problem. Finally, we show that spreading speed k0is continuously dependent on the constants r∞,K,μ,d and satisfiesFinially some discussions and future work will be discussed in Section6.