Results of the diffusive logistic equation in $$\mathcal {H}_{0}^{\varrho ,p}$$

Abstract In the present article, we first introduce the problem of the diffusive logistic equation with memory in Bessel potential spaces, discussing the parameters $$\alpha $$ and $$\widetilde{\eta }$$ and, in particular, their influence on the memory term of the model. Next, we present a result via a lemma that provides an estimate for the integral of a Mittag-Leffler function in terms of the memory effect. Based on this, using the Banach Fixed Point Theorem, Gronwall’s inequality, and the lemma estimating the Mittag-Leffler function, we investigate the existence, uniqueness, regularity, and continuous dependence of weak solutions to the diffusive logistic equation.