A parabolic almost monotonicity formula

We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u +/-( x, s) in the strip S-1 = R-n x (-1, 0] satisfying u +/- >= 0, (Delta - partial derivative(S))u +/- >= -1, u(+) center dot u(-) = 0 in S-1. We also establish a localized version of the formula as well as prove one of its variants. At the end of the paper we give an application to a free boundary problem related to the caloric continuation of heat potentials.