  We analyze the fundamental solution of a time-fractional problem,
establishing existence and uniqueness in an appropriate functional space.
  We also focus on the one-dimensional spatial setting in the case in which the
time-fractional exponent is equal to, or larger than, $\frac12$. In this
situation, we prove that the speed of invasion of the fundamental solution is
at least `almost of square root type', namely it is larger than~$ct^\beta$ for
any given~$c>0$ and~$\beta\in\left(0,\frac12\right)$.
