  In order to prove numerically the global existence and uniqueness of smooth
solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori
upper bounds on the supremum of the numerical range of the linearized operator.
These bounds also have to be easily computable in order to be applicable to our
rigorous a-posteriori methods, as we use them in each time-step of the
numerical discretization. The final goal is to establish global bounds on
smooth local solutions, which then establish global uniqueness.
