Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations

The goal of this work is to study the forward dynamics of positive solutions for the non-autonomous logistic equation u(t) - Delta u = lambda u-b(t)u(p), with p > 1, b(t) > 0, for all t is an element of R, lim(t ->infinity) b(t) = 0. While the pullback asymptotic behaviour for this equation is now well understood, several different possibilities are realized in the forward asymptotic regime