Existence of Non-Negative Solutions for Parabolic Problem on Riemannian Manifold

In this paper, we investigate a perturbed parabolic problem involving the Laplace–Beltrami operator on a smooth compact Riemannian manifold M. For a strongly local Dirichlet form in L2(M). More precisely, we begin by proving that, in the case of the existence of a non-negative solution, the potential can be written as a derivative of some functions which are locally integrable on M; after that, we prove the existence of a non-negative solution for such problems.