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We consider the parabolic evolution differential equations such as heat equation and porus-medium equation on a Riemannian manifold M whose Ricci curvature is bounded below by −(n− 1)k2 and bounded below by 0 on some amount of M . We derive some bounds of differential quantities for a positive solution and some inequalities which resemble Harnack inequalities.

EVOLUTION EQUATIONS ON A RIEMANNIAN MANIFOLD WITH A LOWER RICCI CURVATURE BOUND