Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds

In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation u(t) = Delta u + au log u + bu in M, where a, b are real constants, M is a complete noncompact Riemannian manifold. As a corollary, we give a local gradient estimate for the corresponding elliptic equation: Delta u + au log u + bu = 0 in M, which improves and extends the result of Ma (J Funct Anal 241: 374-382, 2006) and get a bound for the positive solution to this elliptic equation.