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

Abstract

We study the semilinear heat equation ${}^S{D}_0^{\alpha (t)}u=\Delta u+u^p$ on $\RR^n$, where $p>1$ and the Scarpi order changes exponentially from ${\alpha }_1$ to ${\alpha }_2$, with $0<{\alpha }_1,{\alpha }_2<1$. We establish local and maximal mild well-posedness for abstract Scarpi--Volterra equations. For the whole-space heat problem, positive resolvent families yield nonnegative solutions, comparison, a mass identity, and an $L^{\infty }$ blow-up alternative. To study finite-time growth, we use a Gaussian version of Kaplan's weighted-moment method. It reduces the PDE to a scalar nonlinear Volterra inequality and requires no pointwise lower estimate for the non-self-similar Scarpi heat kernel. Consequently, every nontrivial solution blows up in finite time when $1<p<1+2/n$, and sufficiently large data blow up for every $p>1$. For $u_0=A\phi$ with fixed nonzero $0\le \phi \in L^1\cap L^{\infty }$, the maximal lifespan satisfies \[ T_A\asymp A^{-(p-1)/α_1} \qquad(A\to\infty), \] while a subcritical small-amplitude upper bound is governed by the long-time order ${\alpha }_2$. The lifespan bounds are expressed through the inverse of the integrated memory. In the numerical section, we complement these estimates by comparing the Scarpi dynamics with both Caputo endpoint models. Continuous Laplace inversion shows that the logarithmic slope of the integrated memory varies nonmonotonically across the transition. At large amplitudes, the growth thresholds approach those of the Caputo ${\alpha }_1$ model. A nonlinear space--time rescaling probes the long-time regime and reveals nonmonotone threshold-time ratios relative to Caputo ${\alpha }_2$. For fixed-width initial profiles, increasing the transition rate delays the prescribed growth threshold at larger tested amplitudes but advances it at smaller ones.