  We prove certain localized and global differential Harnack inequality for all
positive solutions to the geometric conjugate heat equation coupled to the
forward in time Ricci flow. In this case, the diffusion operator is perturbed
with the curvature operator, precisely, the Laplace-Beltrami operator is
replaced with "$ \Delta - R(x,t)$", where $R$ is the scalar curvature of the
Ricci flow, which is well generalised to the case of nonlinear heat equation
with potential. Our estimates improve on some well known results by weakening
the curvature constraints. As a by product, we obtain some Li-Yau type
differential Harnack estimate. The localized version of our estimate is very
useful in extending the results obtained to noncampact case.
