On The Cauchy Problem For Nonlocal Diffusion Equations

In this paper,we study the spreading properties of the solution of monostable nonlocal diffusion equations.The initial condition u₀ is assumed to be asymptot-ically front-like.When the dispersion kernel J is symmetric and exponentially bounded,we prove that the level sets of the solutionpropagate at a finite prop-agation speed.Furthermore,for exponentially decaying initial data,the propa-gation speeds ofis determined by the decay rate.We find that,if u₀ is a fast exponentially decaying function,then the spatial propagation speed ofat large time is the minimal speed for the existence of traveling wave solutions,andapproaches to the traveling wave solution with the minimal speed.However,if u₀ is a slowly exponentially decaying function,then the spatial propagation speed ofat large time is a constant larger than the minimum speed,andapproaches to a traveling wave solution with this constant.In conclusion:the smaller the decay rates,the larger the propagation speeds of.The initial data which decays exponentially or fast,leads to finite propagation speeds.