This is the abstract of the paper
  "Approximate Solutions to the Abstract Cauchy Problem"
by Boris Baeumer.

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Abstract: We introduce the notion of approximate solutions to deal with
Cauchy problems with  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 R^d is well posed in the approximate (or
regularized) sense if and only if d=1.