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arXiv · 2607.16785

Giga-Kohn-type results for the fully fractional heat equation

Yannick Sire·Juncheng Wei·Ke Wu·Zikai Ye
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Abstract

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.

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Yannick Sire, Juncheng Wei, Ke Wu, Zikai Ye. 2026-07-18. Giga-Kohn-type results for the fully fractional heat equation. https://arxiv.org/abs/2607.16785

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