Local Hessian estimates of solutions to nonlinear parabolic equations along Ricci flow

Abstract

In the paper, we first obtain some Hessian estimates for any positive solution to the nonlinear parabolic equations$ \begin{equation*} \partial_t u(x, t) = \Delta_{g(t)} u(x, t)+\lambda u^{\alpha}(x, t) \end{equation*} $on a Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. Moreover, we only need the curvature tensor is bounded. These estimates imply some local, time reversed Harnack type inequalities. And our conclusion generalizes the Han and Zhang's result, but also our proof process is different and has been slightly simplified. As its applications, we derive the Hessian estimate of harmonic function and eigenfunction.