Asymptotic behavior of solutions to a class of nonlocal non-autonomous diffusion equations

where Omega is a bounded smooth domain in R-N, N >= 1, beta is a positive constant, the coefficient a is a continuous bounded function on R, and K is an integral operator with symmetric kernel. (Ku) (x) := f(RN) J(x,y)u(y)dy, being J a non-negative function continuously differentiable on R-N x R-N and f(RN) J(, y)dy = 1. We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter. is small enough, we show that the origin is locally pullback asymptotically stable. Copyright (C) 2014 John Wiley & Sons, Ltd.