On Nilpotent Semigroups and Solutions with Finite Stopping Time*

We consider here the evolution equation u9 t.sBu t., where B is some unbounded closed operator with dense domain in some separable Hilbert space. We consider the non-trivial classical solution u t. of the last equation such that u t.s0 for t)T. We are interested in finding conditions on operator B for this to occur. There are two cases: in the first case operator B generates a nice semigroup and the inverse to it is an abstract Volterra operator without point spectra, the Cauchy problem is well-posed in this case, and every solution will be zero in finite time; in the second case every point of the complex plane is in the spectral of operator B and so it cannot generate any semigroup and the Cauchy problem in this case is not well-posed. More precisely, there is no uniqueness for solution of the Cauchy problem in the last case. It is interesting to note that such a solution can occur only in two extreme situations: when the spectra of operator B are trivial, or when every point of the complex plane is in it.