Well-posedness and Blow-up in a semilinear heat equation with variable-order Scarpi memory

We study the semilinear heat equation ${}^{S}D_0^{α(t)}u=Δu+u^p$ on $\RR^n$, where $p>1$ and the Scarpi order changes exponentially from $α_1$ to $α_2$, with $0<α_1,α_2<1$. We establish local and maximal mild well-posedness for abstract Scarpi--Volterra equations. For the whole-space heat problem, p