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 nonautonomous attractors of the problem ut = uxx + u 􀀀 (t)u3 when the parameter > 0 varies. Also, we answer a question proposed in [11], concerning the complete description of the structure of the pullback attractor of the problem when 1 < < 4 and, more generally, for ̸= N2, 2 N 2 N. 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.