  The present paper considers the Cauchy-Dirichlet problem for the
time-nonlocal reaction-diffusion equation $$\partial_t
(k\ast(u-u_0))+\mathcal{L}_x [u]=f(u),\,\,\,\, x\in\Omega\subset\mathbb{R}^n,
t>0,$$ where $k\in L^1_{loc}(\mathbb{R}_+),$ $f$ is a locally Lipschitz
function, $\mathcal{L}_x$ is a linear operator. This model arises when studying
the processes of anomalous and ultraslow diffusions. Results regarding the
local and global existence, decay estimates, and blow-up of solutions are
obtained. The obtained results provide partial answers to some open questions
posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016).
Furthermore, possible quasi-linear extensions of the obtained results are
discussed, and some open questions are presented.
