A monotonicity formula for a semilinear fractional parabolic equation

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