Li-Yau estimates for a semilinear parabolic equation on an evolving manifold

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) u^s(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.