  A kind of nonlocal reaction-diffusion equations on an unbounded domain
containing fractional Laplacian operator is analyzed. To be precise, we prove
the convergence of solutions of the equation governed by the fractional
Laplacian to the solutions of the classical equation governed by the standard
Laplacian, when the fractional parameter grows to 1. The existence of global
attractors is investigated as well. The novelty of this paper is concerned with
the convergence of solutions when the fractional parameter varies, which, as
far as the authors are aware, seems to be the first result of this kind of
problems in the literature.
