Abstract
In this paper, we first consider two scalar nonlocal diffusion problems with a free boundary and a fixed boundary. We obtain global existence, uniqueness and longtime behavior of solution of these two problems. The spreading-vanishing dichotomy is established, and some sufficient conditions for spreading and vanishing are also derived. For scalar models, we prove that accelerated spreading could happen if and only if a threshold condition is violated by kernel function. Then we discuss a classical Lotka-Volterra predator-prey model with nonlocal diffusions and a free boundary which can be seen as nonlocal diffusion counterpart of the model in the work of Wang (2014) [24].
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