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

2026/07/18 by Yannick Sire, Juncheng Wei, Ke Wu +1
#math.AP

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Abstract

We consider the semilinear fully fractional heat equation (∂_t-Δ)^σu = |u|^p-1u in ℝⁿ × ℝ_-, 0 < σ< 1. For n≤ 2σ or 1<p≤ (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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