Abstract Degenerate Cauchy Problems in Locally Convex Spaces1

This paper is concerned with the abstract degenerate Cauchy problemŽDCP., ddt BuŽt. AuŽt. Žt 0., BuŽ0. Bu0 , where A and B are closed linear operators in a sequentially complete locally convex space. A C-propagation family forŽDCP. is introducedŽC is an operator., leading to a general C-wellposedness result about ŽDCP.. Moreover, conditions are given ensuring the existence of C-propagation families for thoseŽDCP.with differential operators, on various function spaces with Fr´echet topologies, as coefficient operators. These results are new even in the case of Banach spaces.