HARNACK ESTIMATES FOR THE POROUS MEDIUM EQUATION WITH POTENTIAL UNDER GEOMETRIC FLOW

Let (M, g(t)), t is an element of [0, T) be a closed Riemannian n-manifold whose Riemannian metric g(t) evolves by the geometric flow partial derivative/partial derivative tg(ij )= -2S(ij), where S-ij(t) is a symmetric two-tensor on (M, g(t)). We discuss differential Harnack estimates for positive solution to the porous medium equation with potential, partial derivative u/partial derivative t = Delta u(p) + Su, where S = g(ij) S-ij is the trace of S-ij, on time-dependent Riemannian metric evolving by the above geometric flow.