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As it is well-known, the forwards and pullback dynamics are in generalunrelated. In this paper we present an in-depth study of whether the pullbackattractor is also a forwards attractor for the processes involved with theskew-product semiflow induced by a family of scalar non-autonomousreaction-diffusion equations which are linear in a neighbourhood of zero andhave null upper Lyapunov exponent. Besides, the notion of Li-Yorke chaoticpullback attractor for a process is introduced, and we prove that this chaoticbehaviour might occur for almost all the processes. When the problems areadditionally sublinear, more cases of forwards attraction are found, which hadnot been previously considered even in the case of linear-dissipative ODEs.

Forwards attraction properties in scalar non-autonomous linear dissipative parabolic PDEs. The case of null upper lyapunov exponent