AI 中文总结
研究全分数阶热方程，通过厄米特展开对经典泊松扎耶夫恒等式给出新解释，推广了\(\sigma = 1\)时的单调性公式和刘维尔型定理，还建立了时空非局部单调性公式，为相关研究提供了新方法和结果。
AI 中文摘要
我们考虑半线性全分数阶热方程\[ (\partial_t-\Delta)^\sigma u = |u|^{p - 1}u \quad \text{在 } \mathbb{R}^n \times \mathbb{R}_{-} \text{ 中，} \qquad 0 < \sigma < 1 \]。对于\(n\leq 2\sigma\)或\(1<p\leq \frac{n + 2\sigma}{n - 2\sigma}\)，我们推广了吉加（Giga）和科恩（Kohn）在\(\sigma = 1\)时证明的单调性公式和刘维尔型定理。为克服方程非局部性的困难，我们用厄米特展开对经典的吉加 - 科恩的泊松扎耶夫恒等式给出新解释。我们还为自相似方程建立了时空非局部单调性公式，这是首个不借助斯汀加（Stinga）和托雷亚（Torrea）扩展的时空非局部方程的单调性公式。
英文摘要
We consider the semilinear fully fractional heat equation \[ (\partial_t-Δ)^σu = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < σ< 1. \] For $n\leq 2σ$ or $1<p\leq \frac{n+2σ}{n-2σ}$, we generalize the monotonicity formula and Liouville-type theorem when $σ=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $σ=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.
