We introduce and study a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models in [17] and elsewhere, where "local diffusion" is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [17].

This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.2010 Mathematics Subject Classification. 35K57, 35R20, 92D25. Key words and phrases. SIS model, reaction-diffusion, free boundary, spreading-vanishing dichotomy, nonlocal incidence rate. 247 248 JIA-FENG CAO, WAN-TONG LI AND FEI-YING YANG t −∞ +∞ −∞ K(x−y, t−s)I(y, t)dyds, and obtained the threshold dynamics for the spread of the disease (see [42] for the detailed description about the kernel function K). For more relevant work on the existence of traveling waves of reactiondiffusion equations with nonlocal interaction and time delay, we refer readers to Ducrot et al. [15], Faria et al. [16] and references cited therein. We also refer to Lou [31] for some challenging mathematical problems in evolution of dispersal and population dynamics.Recently, free boundary problems have been studied intensively in many fields. In particular, the well-known Stefan condition has been used to describe the spreading front in many applied problems. For example, it was used to describe the melting of ice in contact with water [35], the wound healing [6], the tumor growth [7] and so on. In order to get a more precise prediction of the location of the spreading front of an invading species, Du et al. [10] firstly studied the spreading-vanishing dichotomy of some invasion species which is described by a diffusive logistic model in the homogenous environment of one dimensional space. Since then, more results for more general free boundary problems have been obtained, for example, see

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