  We study the Cauchy problem for a nonlocal heat equation, which is of
fractional order both in space and time. We prove four main theorems:
  (i) a representation formula for classical solutions,
  (ii) a quantitative decay rate at which the solution tends to the fundamental
solution,
  (iii) optimal $L^2$-decay of mild solutions in all dimensions,
  (iv) $L^2$-decay of weak solutions via energy methods.
  The first result relies on a delicate analysis of the definition of classical
solutions. After proving the representation formula we carefully analyze the
integral representation to obtain the quantitative decay rates of (ii).
  Next we use Fourier analysis techniques to obtain the optimal decay rate for
mild solutions. Here we encounter the critical dimension phenomenon where the
decay rate attains the decay rate of that in a bounded domain for large enough
dimensions. Consequently, the decay rate does not anymore improve when the
dimension increases. The theory is markedly different from that of the standard
caloric functions and this substantially complicates the analysis.
  Finally, we use energy estimates and a comparison principle to prove a
quantitative decay rate for weak solutions defined via a variational
formulation. Our main idea is to show that the $L^2$-norm is actually a
subsolution to a purely time-fractional problem which allows us to use the
known theory to obtain the result.
