If $X$ has the so-called Wiener-regularity, then it generates an analytic semigroup, see the paper by [Arendt and Benilan][1].


  [1]: http://cantor.mathematik.uni-ulm.de/m5/arendt/publications/arendt-pub/short/1999-AreBen-WnrRglHtSmgSpcCntFnc.pdf