《DIFFERENTIAL HARNACK ESTIMATES FOR TIME-DEPENDENT》.pdfVIP

DIFFERENTIAL HARNACK ESTIMATES FOR TIME-DEPENDENT HEAT EQUATIONS WITH POTENTIALS XIAODONG CAO ∗ AND RICHARD S. HAMILTON 8 0 0 2 Abstract. In this paper, we prove a differential Harnack inequality for positive l solutions of time-dependent heat equations with potentials. We also prove a gradient u estimate for the positive solution of the time-dependent heat equation. J 3 1. Introduction ] G In this paper, we will study time-dependent heat equations with potentials on D closed Riemannian manifolds evolving by the Ricci flow . h ∂gij t = −2Rij . a ∂t m We will derive differential Harnack inequalities (also known as Li-Yau type Harnack [ estimates) for positive solutions of parabolic equations of the type 1 ∂f v = △g (t) f + Rf, 8 ∂t 6 where △g (t) depends on time t, R is the scalar curvature of g (t). 5 0 The study of differential Harnack estimates for parabolic equations originated in 7. P. Li and S.-T. Yau’s paper [LY86], in which they proved a differential Harnack 0 inequality for positive solutions of the heat equation on Riemannian manifolds with a 8 0 fixed metric. Namely they proved that, if f is a positive solution to the heat equation : ∂f v = △f i X