Gradient estimates and Harnack inequality for a nonlinear parabolic equation on complete manifolds (Q2254362)

Let \(M\) be an \(n\)-dimensional complete Riemannian manifold with Ricci curvature bounded below. Assume \(0<\alpha <n/(n-1)\) and \(a\) and \(b\) are nonnegative constants. Positive solutions of the nonlinear parabolic equation \(\partial_t u=\Delta u +au \ln u +bu^\alpha \) are studied in \(M\). Gradient estimates of these solutions are given. As a consequence the Harnack inequality is proved.