Singular Stochastic Heat Equations with Fractional Memory

We extend the paracontrolled distribution approach of Gubinelli, Imkeller, and Perkowski to singular stochastic heat equations with Caputo fractional time derivatives. By establishing Schauder estimates for the time-fractional heat flow and controlling history terms, we address the difficulties arising from temporal memory and the lack of the semigroup property. For the one-dimensional rough Burgers-type equation driven by space-time white noise with spatial dissipation $(-\mathrm{\Delta }{)}^{\sigma }$, we prove global well-posedness for $\sigma \in (5\mathrm{/}6,1]$ and $\nu \in (6\sigma \mathrm{/}(12\sigma -5),1]$. For the heat equation with nonlinear multiplicative spatial white noise on the two-dimensional torus, we establish local well-posedness of renormalized solutions and an $L^{\mathrm{\infty }}$ blow-up criterion for every $\nu \in (0,1]$.