  We study the long time behavior of solutions of the non-autonomous
Reaction-Diffusion equation defined on the entire space R^n when external terms
are unbounded in a phase space. The existence of a pullback global attractor
for the equation is established in L^2(R^n) and H^1(R^n), respectively. The
pullback asymptotic compactness of solutions is proved by using uniform a
priori estimates on the tails of solutions outside bounded domains.
