On the upper estimates of fundamental solutions of parabolic equations on Riemannian manifolds (Q1842124)

The authors consider the parabolic equation \[ \Delta u(x,t) + b(x,t) \nabla u(x,t) + h(x,t) u(x,t) - \partial u(x,t)/ \partial t = 0 \tag{1.1} \] on a complete Riemannian manifold \(M\). They derive the gradient estimates and Harnack inequalities for positive solutions of (1.1) and then derive the upper bounds of any positive \(L^2\) fundamental solution of (1.1) when \(h(x,t)\) and \(b(x,t)\) are indepenent of \(t\). Some generalizations of (1.1) are also under consideration. The methods developed earlier by the first author, P. Li and S. T. Yau are applied.