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We prove optimal estimates for the decay in time of solutions to a rathergeneral class of non-local in time subdiffusion equations in $\mathbb{R}^d$. Animportant special case is the time-fractional diffusion equation, which hasseen much interest during the last years, mostly due to its applications in themodeling of anomalous diffusion processes. We follow three different approachesand techniques to study this particular case: (A) estimates based on thefundamental solution and Young's inequality, (B) Fourier multiplier methods,and (C) the energy method. It turns out that the decay behaviour is markedlydifferent from the heat equation case, in particular there occurs a {\emcritical dimension phenomenon}. The general subdiffusion case is treated bymethod (B) and relies on a careful estimation of the underlying relaxationfunction. Several examples of kernels, including the ultraslow diffusion case,illustrate our results.

Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\mathbb{R}^d$