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

Abstract

The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem when the parameter 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 and, more generally, for , . We construct global bounded solutions, 'non-autonomous equilibria', connections between the trivial solution and these 'non-autonomous equilibria' and characterize the -limit and -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.