Abstract
Let (M,g(t)) be a complete Riemannian manifold of dimension n, we obtain Li-Yau type gradient estimates on positive bounded solutions to the following semilinear parabolic equation
 ∂u(t,x) ∕ ∂t = Δ u(t,x) + a(x) us(t,x) -λu(t,x),
 where (t,x) ∈ ([0,T] × M), T < ∞, s > 1, λ ∈ ℝ and a ∈ C2(M) on evolving Riemannian metrics g(t) with bounded below Ricci tensor. The application of our gradient estimates yields the classical differential Harnack inequality, which compares a solution at some time with those at previous time.
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