Generalized parabolic frequency on compact manifolds

In this paper, we first prove monotonicity of a generalized para bolic frequency on weighted closed Riemannian manifolds for some linear heat equation. Secondly, a certain generalized parabolic frequency functional is defined with respect to the solutions of a nonlinear weighted p-heat-type equation on manifolds, and its monotonicity is proved. Notably, the monotonicities are derived with no assumption on both the curvature and the potential function. Further consequences of these monotonicity formulas from which we can get backward uniqueness are discussed