Abstract
This paper addresses the Cauchy-Dirichlet problem in the context of time-nonlocal reaction-diffusion equations, specifically focusing on the 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, and $ \mathcal{L}_x $ is a linear operator. This model is particularly relevant for studying anomalous and ultraslow diffusion processes. Our research contributes to the understanding of this equation by presenting results on local and global existence, decay estimates, and conditions leading to the blow-up of solutions. These findings provide answers to some of the questions raised earlier by Gal and Warma in [C. G. Gal, M. Warma, Springer Nature, Switzerland AG., 2020.] and also by Luchko and Yamamoto in [Y. Luchko, M. Yamamoto, Fract. Calc. Appl. Anal., 19:3 (2016), 676-695]. Additionally, the paper explores potential quasi-linear extensions of these results and outlines several open questions for future research.