  We study the decay/growth rates in all $L^p$ norms of solutions to an
inhomogeneous nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo
$\alpha$-time derivative and a power $\beta$ of the Laplacian when the
dimension is large, $N> 4\beta$. Rates depend strongly on the space-time scale
and on the time behavior of the spatial $L^1$ norm of the forcing term.
