A nonlocal diffusion equation whose solutions develop a free boundary

Let J : R -> R be a nonnegative, smooth compactly supported function such that integral(R) J(r)dr = 1. We consider the nonlocal diffusion problem ut(x, t) = integral(R) J (x - y/u(y,t)) dy - u(x,t) in R x [0, infinity) with a nonnegative initial condition. Under suitable hypotheses we prove existence, uniqueness, as well as the validity of a comparison principle for solutions of this problem. Moreover we show that if u(., 0) is bounded and compactly supported, then u(., t) is compactly supported for all positive times t. This implies the existence of a free boundary, analog to the corresponding one for the porous media equation, for this model.