On Nilpotent Semigroups and Solutions with Finite Stopping Time

Abstract

We consider here the evolution equationu′(t)=Bu(t), whereBis some unbounded closed operator with dense domain in some separable Hilbert space. We consider the non-trivial classical solutionu(t) of the last equation such thatu(t)=0 fort>T. We are interested in finding conditions on operatorBfor this to occur. There are two cases: in the first case operatorBgenerates 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 operatorBand 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 operatorBare trivial, or when every point of the complex plane is in it.