  In this paper we consider a non-local (in time) heat equation on a
time-increasing parabolic set whose boundary is determined by a suitable curve.
We provide a notion of solution for this equation and we study well-posedness
under Dirichlet conditions outside the domain. A maximum principle is proved
and used to derive uniqueness and continuity with respect to the initial datum
of the solutions of the Dirichlet problem. Existence is proved by showing a
stochastic representation based on the delayed Brownian motion killed on the
boundary. Related properties on crossing probabilities of the delayed Brownian
motion are obtained. The asymptotic behaviour of the mean square displacement
of the process is determined, showing that the diffusive behaviour is
anomalous.
