  We prove sharp estimates for the decay in time of solutions to a rather
general class of non-local in time subdiffusion equations on a bounded domain
subject to a homogeneous Dirichlet boundary condition. Important special cases
are the time-fractional and ultraslow diffusion equation, which have seen much
interest during the last years, mostly due to their applications in the
modeling of anomalous diffusion. We study the case where the equation is in
divergence form with bounded measurable coefficients. Our proofs rely on energy
estimates and make use of a new and powerful inequality for
integro-differential operators of the form $\partial_t (k\ast \cdot)$. The
results can be generalized to certain quasilinear equations. We illustrate this
by looking at the time-fractional $p$-Laplace and porous medium equation. Here
it turns out that the decay behaviour is markedly different from that in the
classical parabolic case.
