Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n >= 3. Denote Delta(g) = -div(g)del the Laplace-Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation Delta(g)u(x) + h(x)u(x) = A(x)u(p)(x) + B(x)/u(q)(x) on (M, g). Here, p, q >= 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of ut(x, t) + Delta(g)u(x, t) + h(x) u(x, t) = A(x)u(p)(x, t) + B(x)/u(q)(x, t) on (M, g) with initial data u(x, 0) > 0.