On Nilpotent Semigroups and Solutions with Finite Stopping Time

✍ Scribed by V Schuchman

• Book ID: 102592302
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 165 KB
• Volume: 221
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5102

✦ Synopsis

We consider here the evolution equation uЈ t s Bu t , where B is some unbounded closed operator with dense domain in some separable Hilbert space.

Ž . We consider the non-trivial classical solution u t of the last equation such that Ž . u t s 0 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.