A nonlocal inhomogeneous dispersal process

This article in devoted to the study of the nonlocal dispersal equationut(x,t)=∫RJ(x−yg(y))u(y,t)g(y)dy−u(x,t)in R×[0,∞), and its stationary counterpart. We prove global existence for the initial value problem, and under suitable hypothesis on g and J  , we prove that positive bounded stationary solutions exist. We also analyze the asymptotic behavior of the finite mass solutions as t→∞t→∞, showing that they converge locally to zero.