  As it is well-known, the forwards and pullback dynamics are in general
unrelated. In this paper we present an in-depth study of whether the pullback
attractor is also a forwards attractor for the processes involved with the
skew-product semiflow induced by a family of scalar non-autonomous
reaction-diffusion equations which are linear in a neighbourhood of zero and
have null upper Lyapunov exponent. Besides, the notion of Li-Yorke chaotic
pullback attractor for a process is introduced, and we prove that this chaotic
behaviour might occur for almost all the processes. When the problems are
additionally sublinear, more cases of forwards attraction are found, which had
not been previously considered even in the case of linear-dissipative ODEs.
