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We study the Cauchy problem for a nonlocal heat equation, which is offractional 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 fundamentalsolution,  (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 classicalsolutions. After proving the representation formula we carefully analyze theintegral representation to obtain the quantitative decay rates of (ii).  Next we use Fourier analysis techniques to obtain the optimal decay rate formild solutions. Here we encounter the critical dimension phenomenon where thedecay rate attains the decay rate of that in a bounded domain for large enoughdimensions. Consequently, the decay rate does not anymore improve when thedimension increases. The theory is markedly different from that of the standardcaloric functions and this substantially complicates the analysis.  Finally, we use energy estimates and a comparison principle to prove aquantitative decay rate for weak solutions defined via a variationalformulation. Our main idea is to show that the $L^2$-norm is actually asubsolution to a purely time-fractional problem which allows us to use theknown theory to obtain the result.

Representation of solutions and large-time behavior for fully nonlocal diffusion equations