Well-posedness and Blow-up in a semilinear heat equation with variable-order Scarpi memory
Abstract
We study the semilinear heat equation SD0α(t)u=Δu+up on RRn, where p>1 and the Scarpi order changes exponentially from α1 to α2, with 0<α1,α2<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∞ 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 u0=Aφ with fixed nonzero 0≤φ∈ L1∩ L∞, the maximal lifespan satisfies TAasymp A−(p−1)/α1 (A→∞), while a subcritical small-amplitude upper bound is governed by the long-time order α2. 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 α1 model. A nonlinear space--time rescaling probes the long-time regime and reveals nonmonotone threshold-time ratios relative to Caputo α2. For fixed-width initial profiles, increasing the transition rate delays the prescribed growth threshold at larger tested amplitudes but advances it at smaller ones.
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BibTeX
@misc{yuan2026wellposedness,
title = {{Well-posedness and Blow-up in a semilinear heat equation with variable-order Scarpi memory}},
author = {Pu Yuan and P. A. Zegeling},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16411},
eprint = {2609.16411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16411},
}
RIS
TY - UNPB
TI - Well-posedness and Blow-up in a semilinear heat equation with variable-order Scarpi memory
AU - Pu Yuan
AU - P. A. Zegeling
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16411
UR - https://arxiv.org/abs/2609.16411
AB - We study the semilinear heat equation ^SD_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, positive resolvent families yield nonnegative solutions, comparison, a mass identity, and an L^∞ 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=Aφ with fixed nonzero 0≤φ∈ L^1∩ L^∞, the maximal lifespan satisfies T_Aasymp A^−(p−1)/α_1 (A→∞), while a subcritical small-amplitude upper bound is governed by the long-time order α_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 α_1 model. A nonlinear space--time rescaling probes the long-time regime and reveals nonmonotone threshold-time ratios relative to Caputo α_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.
ER -