AI 中文总结
研究变阶Scarpi记忆半线性热方程的适定性与爆破，通过Kaplan加权矩方法建立有限时间爆破准则，并给出寿命估计与数值验证。
AI 中文摘要
我们研究定义在$\RR^n$上的半线性热方程 ${}^{S}D_0^{\alpha(t)}u=\Delta u+u^p$，其中$p>1$，且Scarpi阶从$\alpha_1$到$\alpha_2$呈指数变化，满足$0<\alpha_1,\alpha_2<1$。我们建立了抽象Scarpi--Volterra方程的局部和极大温和适定性。对于全空间热问题，正预解族产生非负解、比较原理、质量恒等式以及$L^\infty$爆破备选方案。为研究有限时间增长，我们使用Kaplan加权矩方法的高斯版本。该方法将PDE简化为标量非线性Volterra不等式，且不需要对非自相似Scarpi热核进行逐点下界估计。因此，当$1<p<1+2/n$时，每个非平凡解都在有限时间内爆破，且对于所有$p>1$，足够大的数据都会爆破。对于固定的非零$0\le\varphi\in L^1\cap L^\infty$，初值$u_0=A\varphi$的极大寿命满足 $T_A\asymp A^{-(p-1)/\alpha_1} \qquad(A\to\infty)$，而亚临界小振幅上界由长时间阶$\alpha_2$控制。寿命界通过积分记忆的逆函数表示。在数值部分，我们通过将Scarpi动力学与两个Caputo端点模型进行比较来补充这些估计。连续Laplace反演表明，积分记忆的对数斜率在过渡期间非单调变化。在大振幅下，增长阈值接近Caputo $\alpha_1$模型的阈值。非线性时空重缩放探测长时间区域，并揭示相对于Caputo $\alpha_2$的非单调阈值时间比。对于固定宽度的初始轮廓，增加过渡速率在较大测试振幅下延迟规定的增长阈值，但在较小振幅下则提前它。
英文摘要
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, 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 $u_0=Aφ$ with fixed nonzero $0\leφ\in L^1\cap L^\infty$, 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 $α_2$. 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.
