No abstract

In this paper we consider a free boundary problem for a reactiondiffusion logistic equation with a time-dependent growth rate. Such a problem arises in the modeling of information diffusion in online social networks, with the free boundary representing the spreading front of news among users. We present several sharp thresholds for information diffusion that either lasts forever or suspends in finite time. In the former case, we give the asymptotic spreading speed which is determined by a corresponding elliptic equation.

Abstract. Spatial heterogeneity and habitat characteristic are shown to determine the asymptotic profile of the solution to a reaction-diffusion model with free boundary, which describes the moving front of the invasive species. A threshold value R F r 0 (D, t) is introduced to determine the spreading and vanishing of the invasive species. We prove that if R F r 0 (D, t 0 ) ≥ 1 for some t 0 ≥ 0, the spreading must happen; while if R F r 0 (D, 0) < 1, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the diffusion is slow or the occupying habitat is large. In an unfavorable habitat, the species dies out if the initial value of the species is small. However, big initial number of the species is benefit for the species to survive. When the species spreads in the whole habitat, the asymptotic spreading speed is given. Some implications of these theoretical results are also discussed.MSC: primary: 35R35; secondary: 35K60

No abstract

No abstract