A monotonicity formula for a semilinear fractional parabolic equation

Abstract

We apply a high-dimensional parabolic-to-elliptic transformation to the fully fractional, semilinear heat equation $({\partial }_t-\Delta {)}^{s}u=|u{|}^{p-1}u$, where $0<s<1$ and $p>1$, and establish a monotonicity formula for the corresponding extension problem. This result serves as a fractional analogue of the Giga-Kohn monotonicity formula for the local equation ${\partial }_{t}u-\Delta u=|u{|}^{p-1}u.$ The method of proof may be applicable to other nonlinear and nonlocal settings.

Final version