A monotonicity formula for a semilinear fractional parabolic equation

Abstract

By applying a high-dimensional parabolic-to-elliptic transformation, we establish a monotonicity formula for the extension problem of the fractional parabolic semilinear equation $(\partial_t -\Delta)^s u = |u|^{p-1}u$, where $0<s<1$. This is an analogous result to the Giga-Kohn monotonicity formula for the equation $\partial_t u - \Delta u = |u|^{p-1}u.$