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In this paper, certain localized and global gradient estimates for all positive solutions to the geometric heat equation coupled to the Ricci flow either forward or backward in time are proved. As a by product, we obtain various Li-Yau type differential Harnack estimates. We also discuss the case when the diffusion operator is perturbed with the curvature operator (precisely, when the Laplacian is replaced with ”∆−R(x, t)”, R being the scalar operator). This is well generalised to the case of an adjoint heat equation under the Ricci flow. AMS Subject Classification: 35K05, 53C25, 53C44

GRADIENT ESTIMATES FOR HEAT-TYPE EQUATIONS ON MANIFOLDS EVOLVING BY THE RICCI FLOW