A non-autonomous scalar one-dimensional dissipative parabolic problem: the description of the dynamics

The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem u(t) = u(xx) + lambda u - beta(t)u(3) when the parameter lambda > 0 varies. Also, we answer a question proposed in Carvalho et al (2012 Proc. Am. Math. Soc. 140 2357-73), concerning the complete description of the structure of the pullback attractor of the problem when 1 < lambda < 4 and, more generally, for lambda not equal N-2, 2 <= N is an element of N. We construct global bounded solutions, 'non-autonomous equilibria', connections between the trivial solution and these 'non-autonomous equilibria' and characterize the alpha-limit and omega-limit set of global bounded solutions. As a consequence, we show that the global attractor of the associated skew-product flow has a gradient structure. The structure of the related pullback an uniform attractors are derived from that.