  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
