Approximate solutions to the abstract Cauchy problem

We introduce the notion of approximate, continuous solutions to deal with Cauchy problems which have, in general, only distributional or other generalized solutions. We show the equivalence of this notion of solvability with the regularized solutions introduced by I. Cioranescu and G. Lumer. As an example, we show that the backwards heat equation in Omega subset of IRd is wellposed in the approximate (or regularized) sense if and only if d = 1.