Rapid Limit of the Contact Process with Long Range

In this paper, rapid limit of the contact process with long range was studied. By the self duality of the contact process with long range and some estimated formulae about the branching random walk, it was proved that raped limit of the contact process with long range was solution of the following differential equation,$\left\{\begin{array}{l}\frac{\partial U(t, r)}{\partial t}=\frac{1}{2} \Delta U-U(t, r)+\lambda(1-U(t, r)) \int_{r-1}^{r+1} \frac{U(t, x)}{2} d x, \\ U(U, r)=g(r) .\end{array}\right.$.