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.

We study an one-dimensional nonlinear reaction-diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients.We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm-Liouville properties of the solutions of a linear elliptic problem.

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