Properties of the solutions of the conjugate heat equations

In this paper we consider the class ${\cal A}$ of those solutions $u(x,t)$ to the conjugate heat equation ${{\partial}\over{\partial t}}u = -\Delta u + Ru$ on compact K\"ahler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized K\"ahler Ricci flow, blowing up at $T < \infty$), which satisfy Perelman's differential Harnack inequality (6) on $[0,T)$. We show ${\cal A}$ is nonempty. If $|\mathop{\rm Ric}\nolimits(g(t))| \le {{C}\over{T-t}}$, which is always true if we have a type I singularity, we prove the solution $u(x,t)$ satisfies the elliptic type Harnack inequality, with the constants that are uniform in time. If the flow $g(t)$ has a type~I singularity at $T$, then ${\cal A}$ has exactly one element.