The development of the spectral properties for parabolic equations associated with higher-order elliptic operators is quite different from the corresponding second-order operators. In particular, the long time behaviour of solutions can be strikingly different from what one is familiar with for second-order ones. The paper shows that long time growth of the semigroup generated by a fourth-order elliptic operator can occur even for constant coefficients operator. Two different examples of such operators are given. On the other hand also an example of a constant coefficient operator in one dimension which generates a semigroup bounded in time is exhibited. A detailed study of the long time asymptotics in one dimension is made; the generalization of results to higher space dimensions and more complicated operators remains an open problem.