Harnack Inequalities For Simple Heat Equations On Riemannian Manifolds

In this paper,we mainly study Harnack inequalities(the gradient estimates)of possi-tive solutions for two different heat equations by maximum principle.In chapter one,we firstly give a brief introduction of the research background for Harnack inequality(gradient estimate)of heat equation.Next,we introduce the motivation of our problems and mainly results.In chapter two,we mainly introduce the preliminary knowledge of heat equation.We firstly give maximum principle of bounded regions and maximum principle of unbounded regions,and then introduce the maximum principle of general parabolic equations and maximum principles on complete Riemannian manifolds.Finally,we present Bochner formula and its prove.In chapter three,we mainly study nonlinear equation(?)tu=Δu+au log U+Vu,u>0.We firstly study Harnack inequalities on compact Riemannian manifolds,and then expand Harnack inequalities on non-compact Riemannian manifolds.Finally,we considered simi-larly results are showed to be true in case when the manifold(M,g)has compact convex boundary.In chapter four,we study Harnack inequality(gradient estimate)for positive solution to the following linear heat equation on a compact Riemannian manifolds with non-negativeRicci curvature:(?)tu=Δn+ΣWIuI+Vu,where WI and V only depend on the space variable x ∈ M.The novelties of our chapter are the refined global gradient estimates for the corresponding evolution equations.