  This paper deals with a class of initial-boundary value problems for
nonlinear fourth order parabolic systems with time dependent coefficients in a
bounded domain $\Omega \subset \mathbb{R}^N, N\geq 2$. Introducing suitable
conditions on the source terms, we obtain a time interval $[0,T],$ where the
solution remains bounded by deriving a lower bound $T$ of $t^*$. Moreover, we
establish conditions on the shape of the spatial domain and on data sufficient
to guarantee that the solution blows up in finite time $t^*$, deriving an upper
bound for $t^*$.
