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 u t = u xx + λu − β(t)u 3 when the parameter λ > 0 varies. Also, we answer a question proposed in Carvalho et al Proc. Am. Math. Soc. 140 2357, concerning the complete description of the structure of the pullback attractor of the problem when 1 < λ < 4 and, more generally, for λ = N 2 , 2 N ∈ N. We construct global bounded solutions , 'non-autonomous equilibria', connections between the trivial solution and these 'non-autonomous equilibria' and characterize the α-limit and ω-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. London Mathematical Society4 Work partially done while the author visited the the University of São Paulo at São Carlos-SP, the author wishes to thank the Universidade Federal de Lavras for all the support concerning that visit. 5 Research partially supported by CNPq 303929/2015-4, CAPES/DGU 267/2008 and FAPESP 2003/10042-0, Brazil.

A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation

“…Moreover, we also know precisely the diagram of connections between equilibria (see [17]) and that this diagram is stable under autonomous and nonautonomous perturbations (see [18,2,6]). In addition, when we replace b in (1.2) by a time dependent function which is not close to a constant, there has been interesting developments ensuring that the asymptotics still resembles that of (1.2) with b constant (see [11,8]). The introduction of a non-local diffusion changes everything.…”

“…A similar approach, to that described in the above paragraph, has been used to study the inner structure of pullback attractors and uniform attractors (see [8,11]) for a non-autonomous version of (1.2). In order to describe the results for the non-autonomous problem (1.1) we will need to introduce some terminology.…”

“…A notion of non-autonomous equilibria also appears in [16, page 3], associated with random dynamical systems. The notion we have used here is related to the global solutions that play the role of equilibria in the characterization (1.5), see [10,Definition 3.9] for the abstract theoretical situation and [8,11] for (1.1) with a ≡ 1.…”

“…In [8], it is shown that the non-autonomous equilibria play such a role. They are the key factor in the construction of the 'lifted-invariant isolated invariant sets' to which the solutions converge and the rest of the uniform attractor correspond to 'connections' between two of these, see [8] for more details.…”

“…In the case (1.1) with a ≡ 1, the authors in [8] proved results concerning connections between the non-autonomous equilibria. That follows from a careful use of the lap-number property.…”