In this paper, we thought about the positive solutions to the semilinear parabolic equation with three exponents, and obtained several differential Harnack estimates of the positive solutions to the equation. As applications of the main theorems, we found blow-up solutions for the equation and classical Harnack inequalities. Our results generalize some recent works in this direction.

In this paper, we establish some almost-Schur type inequalities on sub-static manifolds, naturally arising in General Relativity. In particular, our results generalize those in ^[1] of Li-Xia for $r$-th mean curvatures of closed sub-static hypersurfaces in space forms and $k$-scalar curvatures for closed locally conformally flat sub-static manifolds. Moreover, our results also generalize those of Cheng ^[2].