  We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$
($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated
with the elliptic operator $\mathscr{L} \coloneqq - \Delta + (-\Delta)^{s}$
($0<s<1$). Firstly, we prove several well-posedness and regularity results of
the associated elliptic and parabolic problems with smooth, and then with
singular boundary-exterior data. Secondly, we show the existence of optimal
solutions of associated optimal control problems, and we characterize the
optimality conditions. This is the first time that such topics have been
presented and studied in a unified fashion for mixed local-nonlocal PDEs with
singular data.
